prompt
listlengths 2
2
| answer
stringlengths 3
48
| src
stringclasses 4
values | messages
listlengths 2
2
|
|---|---|---|---|
[
{
"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 an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \\le a<b<c<d \\le 10$ , and $a+d>b+c$ . How many interesting ordered quadruples are there?",
"type": "text"
}
],
"role": "user"
}
] |
$80$
|
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 an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \\le a<b<c<d \\le 10$ , and $a+d>b+c$ . How many interesting ordered quadruples are there?",
"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 number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\\log(kx)=2\\log(x+2)$ has exactly one real solution.",
"type": "text"
}
],
"role": "user"
}
] |
$501$
|
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 number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\\log(kx)=2\\log(x+2)$ has exactly one real solution.",
"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 $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its elements is $2025$ is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.",
"type": "text"
}
],
"role": "user"
}
] |
$237$
|
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 $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its elements is $2025$ is $\\frac{m}{n}$, 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{}:\nA unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and the length of the rope that is touching the tower is $\\frac{a-\\sqrt{b}}c$ feet, where $a, b,$ and $c$ are positive integers, and $c$ is prime. Find $a+b+c.$",
"type": "text"
}
],
"role": "user"
}
] |
$813$
|
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 unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and the length of the rope that is touching the tower is $\\frac{a-\\sqrt{b}}c$ feet, where $a, b,$ and $c$ are positive integers, and $c$ is prime. 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",
"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 $c$ if $a$ , $b$ , and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$ , where $i^2 = -1$ .",
"type": "text"
}
],
"role": "user"
}
] |
$198$
|
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 $c$ if $a$ , $b$ , and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$ , where $i^2 = -1$ .",
"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 largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.",
"type": "text"
}
],
"role": "user"
}
] |
$486$
|
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 largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive 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 gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$ . ",
"type": "text"
}
],
"role": "user"
}
] |
$106$
|
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 gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. 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 cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$ . [asy] import three; import solids; size(5cm); currentprojection=orthographic(1,-1/6,1/6); draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,360)),white,nolight); triple A =(8*sqrt(6)/3,0,8*sqrt(3)/3), B = (-4*sqrt(6)/3,4*sqrt(2),8*sqrt(3)/3), C = (-4*sqrt(6)/3,-4*sqrt(2),8*sqrt(3)/3), X = (0,0,-2*sqrt(2)); draw(X--X+A--X+A+B--X+A+B+C); draw(X--X+B--X+A+B); draw(X--X+C--X+A+C--X+A+B+C); draw(X+A--X+A+C); draw(X+C--X+C+B--X+A+B+C,linetype(\"2 4\")); draw(X+B--X+C+B,linetype(\"2 4\")); draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,240)),white,nolight); draw((-2,-2*sqrt(3),0)..(4,0,0)..(-2,2*sqrt(3),0)); draw((-4*cos(atan(5)),-4*sin(atan(5)),0)--(-4*cos(atan(5)),-4*sin(atan(5)),-10)..(4,0,-10)..(4*cos(atan(5)),4*sin(atan(5)),-10)--(4*cos(atan(5)),4*sin(atan(5)),0)); draw((-2,-2*sqrt(3),0)..(-4,0,0)..(-2,2*sqrt(3),0),linetype(\"2 4\")); [/asy]",
"type": "text"
}
],
"role": "user"
}
] |
$384$
|
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 cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$ . [asy] import three; import solids; size(5cm); currentprojection=orthographic(1,-1/6,1/6); draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,360)),white,nolight); triple A =(8*sqrt(6)/3,0,8*sqrt(3)/3), B = (-4*sqrt(6)/3,4*sqrt(2),8*sqrt(3)/3), C = (-4*sqrt(6)/3,-4*sqrt(2),8*sqrt(3)/3), X = (0,0,-2*sqrt(2)); draw(X--X+A--X+A+B--X+A+B+C); draw(X--X+B--X+A+B); draw(X--X+C--X+A+C--X+A+B+C); draw(X+A--X+A+C); draw(X+C--X+C+B--X+A+B+C,linetype(\"2 4\")); draw(X+B--X+C+B,linetype(\"2 4\")); draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,240)),white,nolight); draw((-2,-2*sqrt(3),0)..(4,0,0)..(-2,2*sqrt(3),0)); draw((-4*cos(atan(5)),-4*sin(atan(5)),0)--(-4*cos(atan(5)),-4*sin(atan(5)),-10)..(4,0,-10)..(4*cos(atan(5)),4*sin(atan(5)),-10)--(4*cos(atan(5)),4*sin(atan(5)),0)); draw((-2,-2*sqrt(3),0)..(-4,0,0)..(-2,2*sqrt(3),0),linetype(\"2 4\")); [/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": "Find the eighth term of the sequence $1440,$ $1716,$ $1848,\\ldots,$ whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.",
"type": "text"
}
],
"role": "user"
}
] |
$348$
|
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 eighth term of the sequence $1440,$ $1716,$ $1848,\\ldots,$ whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.",
"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 real number $a$ , let $\\lfloor a \\rfloor$ denote the greatest integer less than or equal to $a$ . Let $\\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x,y)$ such that $\\lfloor x \\rfloor ^2 + \\lfloor y \\rfloor ^2 = 25$ . The region $\\mathcal{R}$ is completely contained in a disk of radius $r$ (a disk is the union of a circle and its interior). The minimum value of $r$ can be written as $\\frac {\\sqrt {m}}{n}$ , where $m$ and $n$ are integers and $m$ is not divisible by the square of any prime. Find $m + n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$132$
|
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 real number $a$ , let $\\lfloor a \\rfloor$ denote the greatest integer less than or equal to $a$ . Let $\\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x,y)$ such that $\\lfloor x \\rfloor ^2 + \\lfloor y \\rfloor ^2 = 25$ . The region $\\mathcal{R}$ is completely contained in a disk of radius $r$ (a disk is the union of a circle and its interior). The minimum value of $r$ can be written as $\\frac {\\sqrt {m}}{n}$ , where $m$ and $n$ are integers and $m$ 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",
"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 $\\triangle ABC$ and a point $P$ on one of its sides, call line $\\ell$ the $\\textit{splitting line}$ of $\\triangle ABC$ through $P$ if $\\ell$ passes through $P$ and divides $\\triangle ABC$ into two polygons of equal perimeter. Let $\\triangle ABC$ be a triangle where $BC = 219$ and $AB$ and $AC$ are positive integers. Let $M$ and $N$ be the midpoints of $\\overline{AB}$ and $\\overline{AC},$ respectively, and suppose that the splitting lines of $\\triangle ABC$ through $M$ and $N$ intersect at $30^\\circ.$ Find the perimeter of $\\triangle ABC.$",
"type": "text"
}
],
"role": "user"
}
] |
$459$
|
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 $\\triangle ABC$ and a point $P$ on one of its sides, call line $\\ell$ the $\\textit{splitting line}$ of $\\triangle ABC$ through $P$ if $\\ell$ passes through $P$ and divides $\\triangle ABC$ into two polygons of equal perimeter. Let $\\triangle ABC$ be a triangle where $BC = 219$ and $AB$ and $AC$ are positive integers. Let $M$ and $N$ be the midpoints of $\\overline{AB}$ and $\\overline{AC},$ respectively, and suppose that the splitting lines of $\\triangle ABC$ through $M$ and $N$ intersect at $30^\\circ.$ Find the perimeter of $\\triangle ABC.$",
"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 $\\triangle ABC$ have side lengths $AB=30$ , $BC=32$ , and $AC=34$ . Point $X$ lies in the interior of $\\overline{BC}$ , and points $I_1$ and $I_2$ are the incenters of $\\triangle ABX$ and $\\triangle ACX$ , respectively. Find the minimum possible area of $\\triangle AI_1I_2$ as $X$ varies along $\\overline{BC}$ .",
"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": "Let $\\triangle ABC$ have side lengths $AB=30$ , $BC=32$ , and $AC=34$ . Point $X$ lies in the interior of $\\overline{BC}$ , and points $I_1$ and $I_2$ are the incenters of $\\triangle ABX$ and $\\triangle ACX$ , respectively. Find the minimum possible area of $\\triangle AI_1I_2$ as $X$ varies along $\\overline{BC}$ .",
"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 real numbers $s$ , let $\\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$ . The set of all $s$ for which $\\tau(s)$ is nonempty, but all triangles in $\\tau(s)$ are congruent, is an interval $[a,b)$ . Find $a^2+b^2$ .",
"type": "text"
}
],
"role": "user"
}
] |
$736$
|
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 real numbers $s$ , let $\\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$ . The set of all $s$ for which $\\tau(s)$ is nonempty, but all triangles in $\\tau(s)$ are congruent, is an interval $[a,b)$ . Find $a^2+b^2$ .",
"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 $A_1, A_2, A_3, \\ldots, A_{12}$ be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set $\\{A_1,A_2,A_3,\\ldots,A_{12}\\}$ ?",
"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{}:\nLet $A_1, A_2, A_3, \\ldots, A_{12}$ be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set $\\{A_1,A_2,A_3,\\ldots,A_{12}\\}$ ?",
"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": "Call a permutation $a_1, a_2, \\ldots, a_n$ of the integers $1, 2, \\ldots, n$ quasi-increasing if $a_k \\leq a_{k+1} + 2$ for each $1 \\leq k \\leq n-1$ . For example, $53421$ and $14253$ are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$ , but $45123$ is not. Find the number of quasi-increasing permutations of the integers $1, 2, \\ldots, 7$ .",
"type": "text"
}
],
"role": "user"
}
] |
$486$
|
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": "Call a permutation $a_1, a_2, \\ldots, a_n$ of the integers $1, 2, \\ldots, n$ quasi-increasing if $a_k \\leq a_{k+1} + 2$ for each $1 \\leq k \\leq n-1$ . For example, $53421$ and $14253$ are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$ , but $45123$ is not. Find the number of quasi-increasing permutations of the integers $1, 2, \\ldots, 7$ .",
"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",
"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 function f is defined on the set of integers and satisfies $f(n)= \\begin{cases} n-3 & \\mbox{if }n\\ge 1000 \\\\ f(f(n+5)) & \\mbox{if }n<1000 \\end{cases}$ Find $f(84)$ .",
"type": "text"
}
],
"role": "user"
}
] |
$997$
|
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 function f is defined on the set of integers and satisfies $f(n)= \\begin{cases} n-3 & \\mbox{if }n\\ge 1000 \\\\ f(f(n+5)) & \\mbox{if }n<1000 \\end{cases}$ Find $f(84)$ .",
"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 each positive integer $n,$ let $f(n) = \\sum_{k = 1}^{100} \\lfloor \\log_{10} (kn) \\rfloor$ . Find the largest value of $n$ for which $f(n) \\le 300$ . Note: $\\lfloor x \\rfloor$ is the greatest integer less than or equal to $x$ .",
"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",
"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 each positive integer $n,$ let $f(n) = \\sum_{k = 1}^{100} \\lfloor \\log_{10} (kn) \\rfloor$ . Find the largest value of $n$ for which $f(n) \\le 300$ . Note: $\\lfloor x \\rfloor$ is the greatest integer 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\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 integer \\[N = 9 + 99 + 999 + 9999 + \\cdots + \\underbrace{99\\ldots 99}_\\text{321 digits}.\\] Find the sum of the digits of $N$ .",
"type": "text"
}
],
"role": "user"
}
] |
$342$
|
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 integer \\[N = 9 + 99 + 999 + 9999 + \\cdots + \\underbrace{99\\ldots 99}_\\text{321 digits}.\\] Find the sum of the 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{}:\nFind the number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}.$",
"type": "text"
}
],
"role": "user"
}
] |
$435$
|
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 number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}.$",
"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{}:\nLet $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$ , $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$",
"type": "text"
}
],
"role": "user"
}
] |
$676$
|
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 polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$ , $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$",
"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 points $A$ , $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$ . It is given that $AB=13$ , $BC=14$ , $CA=15$ , and that the distance from $O$ to triangle $ABC$ is $\\frac{m\\sqrt{n}}k$ , where $m$ , $n$ , and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k$ .",
"type": "text"
}
],
"role": "user"
}
] |
$118$
|
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 points $A$ , $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$ . It is given that $AB=13$ , $BC=14$ , $CA=15$ , and that the distance from $O$ to triangle $ABC$ is $\\frac{m\\sqrt{n}}k$ , where $m$ , $n$ , and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k$ .",
"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": "Charles has two six-sided dice. One of the die is fair, and the other die is biased so that it comes up six with probability $\\frac{2}{3}$ and each of the other five sides has probability $\\frac{1}{15}$ . Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six is $\\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .",
"type": "text"
}
],
"role": "user"
}
] |
$167$
|
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": "Charles has two six-sided dice. One of the die is fair, and the other die is biased so that it comes up six with probability $\\frac{2}{3}$ and each of the other five sides has probability $\\frac{1}{15}$ . Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six 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\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 greatest integer multiple of 8, whose digits are all different. What is the remainder when $N$ is divided by 1000?",
"type": "text"
}
],
"role": "user"
}
] |
$120$
|
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 greatest integer multiple of 8, whose digits are all different. What is 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{}:\nDefine $\\operatorname{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0$. Compute\n\n$$\n\\sum_{n=1}^{\\infty} \\frac{\\operatorname{sgn}\\left(\\sin \\left(2^{n}\\right)\\right)}{2^{n}}\n$$\n(The arguments to sin are in radians.)",
"type": "text"
}
],
"role": "user"
}
] |
$1-\frac{2}{\pi}$
|
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{}:\nDefine $\\operatorname{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0$. Compute\n\n$$\n\\sum_{n=1}^{\\infty} \\frac{\\operatorname{sgn}\\left(\\sin \\left(2^{n}\\right)\\right)}{2^{n}}\n$$\n(The arguments to sin are in radians.)",
"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 sets $A = \\{z : z^{18} = 1\\}$ and $B = \\{w : w^{48} = 1\\}$ are both sets of complex roots of unity. The set $C = \\{zw : z \\in A ~ \\mbox{and} ~ w \\in B\\}$ is also a set of complex roots of unity. How many distinct elements are in $C^{}_{}$ ?",
"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\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 sets $A = \\{z : z^{18} = 1\\}$ and $B = \\{w : w^{48} = 1\\}$ are both sets of complex roots of unity. The set $C = \\{zw : z \\in A ~ \\mbox{and} ~ w \\in B\\}$ is also a set of complex roots of unity. How many distinct elements are in $C^{}_{}$ ?",
"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",
"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 $p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3$ . Suppose that $p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1) = p(1,1) = p(1, - 1) = p(2,2) = 0$ . There is a point $\\left(\\frac {a}{c},\\frac {b}{c}\\right)$ for which $p\\left(\\frac {a}{c},\\frac {b}{c}\\right) = 0$ for all such polynomials, where $a$ , $b$ , and $c$ are positive integers, $a$ and $c$ are relatively prime, and $c > 1$ . Find $a + b + c$ .",
"type": "text"
}
],
"role": "user"
}
] |
$40$
|
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 $p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3$ . Suppose that $p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1) = p(1,1) = p(1, - 1) = p(2,2) = 0$ . There is a point $\\left(\\frac {a}{c},\\frac {b}{c}\\right)$ for which $p\\left(\\frac {a}{c},\\frac {b}{c}\\right) = 0$ for all such polynomials, where $a$ , $b$ , and $c$ are positive integers, $a$ and $c$ are relatively prime, and $c > 1$ . 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": "Find the least positive integer $n$ such that when $3^n$ is written in base $143$ , its two right-most digits in base $143$ are $01$ .",
"type": "text"
}
],
"role": "user"
}
] |
$195$
|
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 least positive integer $n$ such that when $3^n$ is written in base $143$ , its two right-most digits in base $143$ are $01$ .",
"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 that $x, y$, and $z$ are positive real numbers such that\n\n$$\nx^{\\log _{2}(y z)}=2^{8} \\cdot 3^{4}, \\quad y^{\\log _{2}(z x)}=2^{9} \\cdot 3^{6}, \\quad \\text { and } \\quad z^{\\log _{2}(x y)}=2^{5} \\cdot 3^{10}\n$$\ncompute the smallest possible value of $x y z$.",
"type": "text"
}
],
"role": "user"
}
] |
$\frac{1}{576}$
|
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": "Given that $x, y$, and $z$ are positive real numbers such that\n\n$$\nx^{\\log _{2}(y z)}=2^{8} \\cdot 3^{4}, \\quad y^{\\log _{2}(z x)}=2^{9} \\cdot 3^{6}, \\quad \\text { and } \\quad z^{\\log _{2}(x y)}=2^{5} \\cdot 3^{10}\n$$\ncompute the smallest possible value of $x y z$.",
"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": "The repeating decimals $0.abab\\overline{ab}$ and $0.abcabc\\overline{abc}$ satisfy $0.abab\\overline{ab}+0.abcabc\\overline{abc}=\\frac{33}{37},$ where $a$ , $b$ , and $c$ are (not necessarily distinct) digits. Find the three digit number $abc$ .",
"type": "text"
}
],
"role": "user"
}
] |
$447$
|
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 repeating decimals $0.abab\\overline{ab}$ and $0.abcabc\\overline{abc}$ satisfy $0.abab\\overline{ab}+0.abcabc\\overline{abc}=\\frac{33}{37},$ where $a$ , $b$ , and $c$ are (not necessarily distinct) digits. Find the three digit number $abc$ .",
"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 the integer $k^{}_{}$ is added to each of the numbers $36^{}_{}$ , $300^{}_{}$ , and $596^{}_{}$ , one obtains the squares of three consecutive terms of an arithmetic series. Find $k^{}_{}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$925$
|
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 the integer $k^{}_{}$ is added to each of the numbers $36^{}_{}$ , $300^{}_{}$ , and $596^{}_{}$ , one obtains the squares of three consecutive terms of an arithmetic series. Find $k^{}_{}$ .",
"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": "In $\\triangle{ABC}$ with $AB = 12$ , $BC = 13$ , and $AC = 15$ , let $M$ be a point on $\\overline{AC}$ such that the incircles of $\\triangle{ABM}$ and $\\triangle{BCM}$ have equal radii. Then $\\frac{AM}{CM} = \\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$ .",
"type": "text"
}
],
"role": "user"
}
] |
$45$
|
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": "In $\\triangle{ABC}$ with $AB = 12$ , $BC = 13$ , and $AC = 15$ , let $M$ be a point on $\\overline{AC}$ such that the incircles of $\\triangle{ABM}$ and $\\triangle{BCM}$ have equal radii. Then $\\frac{AM}{CM} = \\frac{p}{q}$ , 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\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 stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table. It is found that, reading from left to right, the labels on the cards are now in ascending order: $1,2,3,\\ldots,1999,2000.$ In the original stack of cards, how many cards were above the card labeled $1999$ ?",
"type": "text"
}
],
"role": "user"
}
] |
$927$
|
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 stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table. It is found that, reading from left to right, the labels on the cards are now in ascending order: $1,2,3,\\ldots,1999,2000.$ In the original stack of cards, how many cards were above the card labeled $1999$ ?",
"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": "In isosceles trapezoid $ABCD,$ parallel bases $\\overline{AB}$ and $\\overline{CD}$ have lengths $500$ and $650,$ respectively, and $AD=BC=333.$ The angle bisectors of $\\angle A$ and $\\angle D$ meet at $P,$ and the angle bisectors of $\\angle B$ and $\\angle C$ meet at $Q.$ Find $PQ.$",
"type": "text"
}
],
"role": "user"
}
] |
$242$
|
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": "In isosceles trapezoid $ABCD,$ parallel bases $\\overline{AB}$ and $\\overline{CD}$ have lengths $500$ and $650,$ respectively, and $AD=BC=333.$ The angle bisectors of $\\angle A$ and $\\angle D$ meet at $P,$ and the angle bisectors of $\\angle B$ and $\\angle C$ meet at $Q.$ Find $PQ.$",
"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 $ABCD$ be a convex quadrilateral with $AB=2, AD=7,$ and $CD=3$ such that the bisectors of acute angles $\\angle{DAB}$ and $\\angle{ADC}$ intersect at the midpoint of $\\overline{BC}.$ Find the square of the area of $ABCD.$",
"type": "text"
}
],
"role": "user"
}
] |
$180$
|
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 $ABCD$ be a convex quadrilateral with $AB=2, AD=7,$ and $CD=3$ such that the bisectors of acute angles $\\angle{DAB}$ and $\\angle{ADC}$ intersect at the midpoint of $\\overline{BC}.$ Find the square of the area of $ABCD.$",
"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": "Define a sequence recursively by $f_1(x)=|x-1|$ and $f_n(x)=f_{n-1}(|x-n|)$ for integers $n>1$ . Find the least value of $n$ such that the sum of the zeros of $f_n$ exceeds $500,000$ .",
"type": "text"
}
],
"role": "user"
}
] |
$101$
|
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": "Define a sequence recursively by $f_1(x)=|x-1|$ and $f_n(x)=f_{n-1}(|x-n|)$ for integers $n>1$ . Find the least value of $n$ such that the sum of the zeros of $f_n$ exceeds $500,000$ .",
"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 complex coefficients whose $x^2$ coefficient is $1.$ Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b.$ Find the sum of all possible values of $(a+b)^2.$",
"type": "text"
}
],
"role": "user"
}
] |
$85$
|
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 complex coefficients whose $x^2$ coefficient is $1.$ Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b.$ Find the sum of all possible values of $(a+b)^2.$",
"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": "Eight circles of diameter 1 are packed in the first quadrant of the coordinate plane as shown. Let region $\\mathcal{R}$ be the union of the eight circular regions. Line $l,$ with slope 3, divides $\\mathcal{R}$ into two regions of equal area. Line $l$ 's equation can be expressed in the form $ax=by+c,$ where $a, b,$ and $c$ are positive integers whose greatest common divisor is 1. Find $a^2+b^2+c^2.$ [asy] unitsize(0.50cm); draw((0,-1)--(0,6)); draw((-1,0)--(6,0)); draw(shift(1,1)*unitcircle); draw(shift(1,3)*unitcircle); draw(shift(1,5)*unitcircle); draw(shift(3,1)*unitcircle); draw(shift(3,3)*unitcircle); draw(shift(3,5)*unitcircle); draw(shift(5,1)*unitcircle); draw(shift(5,3)*unitcircle); [/asy]",
"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\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": "Eight circles of diameter 1 are packed in the first quadrant of the coordinate plane as shown. Let region $\\mathcal{R}$ be the union of the eight circular regions. Line $l,$ with slope 3, divides $\\mathcal{R}$ into two regions of equal area. Line $l$ 's equation can be expressed in the form $ax=by+c,$ where $a, b,$ and $c$ are positive integers whose greatest common divisor is 1. Find $a^2+b^2+c^2.$ [asy] unitsize(0.50cm); draw((0,-1)--(0,6)); draw((-1,0)--(6,0)); draw(shift(1,1)*unitcircle); draw(shift(1,3)*unitcircle); draw(shift(1,5)*unitcircle); draw(shift(3,1)*unitcircle); draw(shift(3,3)*unitcircle); draw(shift(3,5)*unitcircle); draw(shift(5,1)*unitcircle); draw(shift(5,3)*unitcircle); [/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": "Trapezoid $ABCD^{}_{}$ has sides $AB=92^{}_{}$ , $BC=50^{}_{}$ , $CD=19^{}_{}$ , and $AD=70^{}_{}$ , with $AB^{}_{}$ parallel to $CD^{}_{}$ . A circle with center $P^{}_{}$ on $AB^{}_{}$ is drawn tangent to $BC^{}_{}$ and $AD^{}_{}$ . Given that $AP^{}_{}=\\frac mn$ , where $m^{}_{}$ and $n^{}_{}$ are relatively prime positive integers, find $m+n^{}_{}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$164$
|
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": "Trapezoid $ABCD^{}_{}$ has sides $AB=92^{}_{}$ , $BC=50^{}_{}$ , $CD=19^{}_{}$ , and $AD=70^{}_{}$ , with $AB^{}_{}$ parallel to $CD^{}_{}$ . A circle with center $P^{}_{}$ on $AB^{}_{}$ is drawn tangent to $BC^{}_{}$ and $AD^{}_{}$ . Given that $AP^{}_{}=\\frac mn$ , 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",
"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 $\\triangle ABC$ , $AC = BC$ , and point $D$ is on $\\overline{BC}$ so that $CD = 3\\cdot BD$ . Let $E$ be the midpoint of $\\overline{AD}$ . Given that $CE = \\sqrt{7}$ and $BE = 3$ , the area of $\\triangle ABC$ 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"
}
] |
$10$
|
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{}:\nIn $\\triangle ABC$ , $AC = BC$ , and point $D$ is on $\\overline{BC}$ so that $CD = 3\\cdot BD$ . Let $E$ be the midpoint of $\\overline{AD}$ . Given that $CE = \\sqrt{7}$ and $BE = 3$ , the area of $\\triangle ABC$ 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": "Find the integer that is closest to $1000\\sum_{n=3}^{10000}\\frac1{n^2-4}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$521$
|
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 integer that is closest to $1000\\sum_{n=3}^{10000}\\frac1{n^2-4}$ .",
"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 twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB$, $CJ$, $DG$, $EK$, $FL$, $HI$. The probability that the last word listed contains $G$ is $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.",
"type": "text"
}
],
"role": "user"
}
] |
$821$
|
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{}:\nThe twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB$, $CJ$, $DG$, $EK$, $FL$, $HI$. The probability that the last word listed contains $G$ is $\\frac mn$, 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": "In $\\triangle ABC, AB = 3, BC = 4,$ and $CA = 5$ . Circle $\\omega$ intersects $\\overline{AB}$ at $E$ and $B, \\overline{BC}$ at $B$ and $D,$ and $\\overline{AC}$ at $F$ and $G$ . Given that $EF=DF$ and $\\frac{DG}{EG} = \\frac{3}{4},$ length $DE=\\frac{a\\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$ .",
"type": "text"
}
],
"role": "user"
}
] |
$41$
|
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": "In $\\triangle ABC, AB = 3, BC = 4,$ and $CA = 5$ . Circle $\\omega$ intersects $\\overline{AB}$ at $E$ and $B, \\overline{BC}$ at $B$ and $D,$ and $\\overline{AC}$ at $F$ and $G$ . Given that $EF=DF$ and $\\frac{DG}{EG} = \\frac{3}{4},$ length $DE=\\frac{a\\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. 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",
"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 $a_2+a_4+a_6+a_8+\\ldots+a_{98}$ if $a_1$ , $a_2$ , $a_3\\ldots$ is an arithmetic progression with common difference 1, and $a_1+a_2+a_3+\\ldots+a_{98}=137$ .",
"type": "text"
}
],
"role": "user"
}
] |
$93$
|
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 $a_2+a_4+a_6+a_8+\\ldots+a_{98}$ if $a_1$ , $a_2$ , $a_3\\ldots$ is an arithmetic progression with common difference 1, and $a_1+a_2+a_3+\\ldots+a_{98}=137$ .",
"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": "Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the $50$ -mile mark at exactly the same time. How many minutes has it taken them?",
"type": "text"
}
],
"role": "user"
}
] |
$620$
|
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": "Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the $50$ -mile mark at exactly the same time. How many minutes has it taken them?",
"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 all positive integers $x$ , let \\[f(x)=\\begin{cases}1 &\\mbox{if }x = 1\\\\ \\frac x{10} &\\mbox{if }x\\mbox{ is divisible by 10}\\\\ x+1 &\\mbox{otherwise}\\end{cases}\\] and define a sequence as follows: $x_1=x$ and $x_{n+1}=f(x_n)$ for all positive integers $n$ . Let $d(x)$ be the smallest $n$ such that $x_n=1$ . (For example, $d(100)=3$ and $d(87)=7$ .) Let $m$ be the number of positive integers $x$ such that $d(x)=20$ . Find the sum of the distinct prime factors of $m$ .",
"type": "text"
}
],
"role": "user"
}
] |
$511$
|
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 all positive integers $x$ , let \\[f(x)=\\begin{cases}1 &\\mbox{if }x = 1\\\\ \\frac x{10} &\\mbox{if }x\\mbox{ is divisible by 10}\\\\ x+1 &\\mbox{otherwise}\\end{cases}\\] and define a sequence as follows: $x_1=x$ and $x_{n+1}=f(x_n)$ for all positive integers $n$ . Let $d(x)$ be the smallest $n$ such that $x_n=1$ . (For example, $d(100)=3$ and $d(87)=7$ .) Let $m$ be the number of positive integers $x$ such that $d(x)=20$ . Find the sum of the distinct prime factors of $m$ .",
"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{}:\nBeatrix is going to place six rooks on a $6 \\times 6$ chessboard where both the rows and columns are labeled $1$ to $6$ ; the rooks are placed so that no two rooks are in the same row or the same column. The value of a square is the sum of its row number and column number. The score of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is $\\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .",
"type": "text"
}
],
"role": "user"
}
] |
$371$
|
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{}:\nBeatrix is going to place six rooks on a $6 \\times 6$ chessboard where both the rows and columns are labeled $1$ to $6$ ; the rooks are placed so that no two rooks are in the same row or the same column. The value of a square is the sum of its row number and column number. The score of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations 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{}:\nTrapezoid $A B C D$, with $A B \\| C D$, has side lengths $A B=11, B C=8, C D=19$, and $D A=4$. Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\\triangle A B C, \\triangle B C D$, $\\triangle C D A$, and $\\triangle D A B$.",
"type": "text"
}
],
"role": "user"
}
] |
$9\sqrt{15}$
|
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{}:\nTrapezoid $A B C D$, with $A B \\| C D$, has side lengths $A B=11, B C=8, C D=19$, and $D A=4$. Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\\triangle A B C, \\triangle B C D$, $\\triangle C D A$, and $\\triangle D A B$.",
"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\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 each ordered pair of real numbers $(x,y)$ satisfying \\[\\log_2(2x+y) = \\log_4(x^2+xy+7y^2)\\] there is a real number $K$ such that \\[\\log_3(3x+y) = \\log_9(3x^2+4xy+Ky^2).\\] Find the product of all possible values of $K$ .",
"type": "text"
}
],
"role": "user"
}
] |
$189$
|
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 each ordered pair of real numbers $(x,y)$ satisfying \\[\\log_2(2x+y) = \\log_4(x^2+xy+7y^2)\\] there is a real number $K$ such that \\[\\log_3(3x+y) = \\log_9(3x^2+4xy+Ky^2).\\] Find the product of all possible values 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{}:\nEquilateral triangle $\\triangle ABC$ is inscribed in circle $\\omega$ with radius $18.$ Circle $\\omega_A$ is tangent to sides $\\overline{AB}$ and $\\overline{AC}$ and is internally tangent to $\\omega.$ Circles $\\omega_B$ and $\\omega_C$ are defined analogously. Circles $\\omega_A,$ $\\omega_B,$ and $\\omega_C$ meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of $\\triangle ABC$ are the vertices of a large equilateral triangle in the interior of $\\triangle ABC,$ and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of $\\triangle ABC.$ The side length of the smaller equilateral triangle can be written as $\\sqrt{a} - \\sqrt{b},$ where $a$ and $b$ are positive integers. Find $a+b.$",
"type": "text"
}
],
"role": "user"
}
] |
$378$
|
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{}:\nEquilateral triangle $\\triangle ABC$ is inscribed in circle $\\omega$ with radius $18.$ Circle $\\omega_A$ is tangent to sides $\\overline{AB}$ and $\\overline{AC}$ and is internally tangent to $\\omega.$ Circles $\\omega_B$ and $\\omega_C$ are defined analogously. Circles $\\omega_A,$ $\\omega_B,$ and $\\omega_C$ meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of $\\triangle ABC$ are the vertices of a large equilateral triangle in the interior of $\\triangle ABC,$ and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of $\\triangle ABC.$ The side length of the smaller equilateral triangle can be written as $\\sqrt{a} - \\sqrt{b},$ where $a$ and $b$ are positive integers. Find $a+b.$",
"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\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 all polynomials of the form $z^3 + az^2 + bz + c$ , where $a$ , $b$ , and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$ .",
"type": "text"
}
],
"role": "user"
}
] |
$540$
|
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 all polynomials of the form $z^3 + az^2 + bz + c$ , where $a$ , $b$ , and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$ .",
"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 Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below. \\[\\begin{array}{c@{\\hspace{8em}} c@{\\hspace{6pt}}c@{\\hspace{6pt}}c@{\\hspace{6pt}}c@{\\hspace{4pt}}c@{\\hspace{2pt}} c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{3pt}}c@{\\hspace{6pt}} c@{\\hspace{6pt}}c@{\\hspace{6pt}}c} \\vspace{4pt} \\text{Row 0: } & & & & & & & 1 & & & & & & \\\\\\vspace{4pt} \\text{Row 1: } & & & & & & 1 & & 1 & & & & & \\\\\\vspace{4pt} \\text{Row 2: } & & & & & 1 & & 2 & & 1 & & & & \\\\\\vspace{4pt} \\text{Row 3: } & & & & 1 & & 3 & & 3 & & 1 & & & \\\\\\vspace{4pt} \\text{Row 4: } & & & 1 & & 4 & & 6 & & 4 & & 1 & & \\\\\\vspace{4pt} \\text{Row 5: } & & 1 & & 5 & &10& &10 & & 5 & & 1 & \\\\\\vspace{4pt} \\text{Row 6: } & 1 & & 6 & &15& &20& &15 & & 6 & & 1 \\end{array}\\] In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$ ?",
"type": "text"
}
],
"role": "user"
}
] |
$62$
|
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 Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below. \\[\\begin{array}{c@{\\hspace{8em}} c@{\\hspace{6pt}}c@{\\hspace{6pt}}c@{\\hspace{6pt}}c@{\\hspace{4pt}}c@{\\hspace{2pt}} c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{3pt}}c@{\\hspace{6pt}} c@{\\hspace{6pt}}c@{\\hspace{6pt}}c} \\vspace{4pt} \\text{Row 0: } & & & & & & & 1 & & & & & & \\\\\\vspace{4pt} \\text{Row 1: } & & & & & & 1 & & 1 & & & & & \\\\\\vspace{4pt} \\text{Row 2: } & & & & & 1 & & 2 & & 1 & & & & \\\\\\vspace{4pt} \\text{Row 3: } & & & & 1 & & 3 & & 3 & & 1 & & & \\\\\\vspace{4pt} \\text{Row 4: } & & & 1 & & 4 & & 6 & & 4 & & 1 & & \\\\\\vspace{4pt} \\text{Row 5: } & & 1 & & 5 & &10& &10 & & 5 & & 1 & \\\\\\vspace{4pt} \\text{Row 6: } & 1 & & 6 & &15& &20& &15 & & 6 & & 1 \\end{array}\\] In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$ ?",
"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 nonnegative integers $a$ and $b$ with $a + b \\leq 6$ , let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$ . Let $S$ denote the sum of all $T(a, b)$ , where $a$ and $b$ are nonnegative integers with $a + b \\leq 6$ . Find the remainder when $S$ is divided by $1000$ .",
"type": "text"
}
],
"role": "user"
}
] |
$564$
|
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 nonnegative integers $a$ and $b$ with $a + b \\leq 6$ , let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$ . Let $S$ denote the sum of all $T(a, b)$ , where $a$ and $b$ are nonnegative integers with $a + b \\leq 6$ . Find the remainder when $S$ 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": "Given a nonnegative real number $x$ , let $\\langle x\\rangle$ denote the fractional part of $x$ ; that is, $\\langle x\\rangle=x-\\lfloor x\\rfloor$ , where $\\lfloor x\\rfloor$ denotes the greatest integer less than or equal to $x$ . Suppose that $a$ is positive, $\\langle a^{-1}\\rangle=\\langle a^2\\rangle$ , and $2<a^2<3$ . Find the value of $a^{12}-144a^{-1}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$233$
|
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 a nonnegative real number $x$ , let $\\langle x\\rangle$ denote the fractional part of $x$ ; that is, $\\langle x\\rangle=x-\\lfloor x\\rfloor$ , where $\\lfloor x\\rfloor$ denotes the greatest integer less than or equal to $x$ . Suppose that $a$ is positive, $\\langle a^{-1}\\rangle=\\langle a^2\\rangle$ , and $2<a^2<3$ . Find the value of $a^{12}-144a^{-1}$ .",
"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 number of pairs $(m,n)$ of positive integers with $1\\le m<n\\le 30$ such that there exists a real number $x$ satisfying \\[\\sin(mx)+\\sin(nx)=2.\\]",
"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\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 number of pairs $(m,n)$ of positive integers with $1\\le m<n\\le 30$ such that there exists a real number $x$ satisfying \\[\\sin(mx)+\\sin(nx)=2.\\]",
"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 $EFGH$ , $EFDC$ , and $EHBC$ be three adjacent square faces of a cube, for which $EC = 8$ , and let $A$ be the eighth vertex of the cube. Let $I$ , $J$ , and $K$ , be the points on $\\overline{EF}$ , $\\overline{EH}$ , and $\\overline{EC}$ , respectively, so that $EI = EJ = EK = 2$ . A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\\overline{AE}$ , and containing the edges, $\\overline{IJ}$ , $\\overline{JK}$ , and $\\overline{KI}$ . The surface area of $S$ , including the walls of the tunnel, is $m + n\\sqrt {p}$ , where $m$ , $n$ , and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m + n + p$ .",
"type": "text"
}
],
"role": "user"
}
] |
$417$
|
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 $EFGH$ , $EFDC$ , and $EHBC$ be three adjacent square faces of a cube, for which $EC = 8$ , and let $A$ be the eighth vertex of the cube. Let $I$ , $J$ , and $K$ , be the points on $\\overline{EF}$ , $\\overline{EH}$ , and $\\overline{EC}$ , respectively, so that $EI = EJ = EK = 2$ . A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\\overline{AE}$ , and containing the edges, $\\overline{IJ}$ , $\\overline{JK}$ , and $\\overline{KI}$ . The surface area of $S$ , including the walls of the tunnel, is $m + n\\sqrt {p}$ , where $m$ , $n$ , and $p$ are positive integers 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\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": "Ellina has twelve blocks, two each of red ( $\\textbf{R}$ ), blue ( $\\textbf{B}$ ), yellow ( $\\textbf{Y}$ ), green ( $\\textbf{G}$ ), orange ( $\\textbf{O}$ ), and purple ( $\\textbf{P}$ ). Call an arrangement of blocks $\\textit{even}$ if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement \\[\\textbf{R B B Y G G Y R O P P O}\\] is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is $\\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ ",
"type": "text"
}
],
"role": "user"
}
] |
$247$
|
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": "Ellina has twelve blocks, two each of red ( $\\textbf{R}$ ), blue ( $\\textbf{B}$ ), yellow ( $\\textbf{Y}$ ), green ( $\\textbf{G}$ ), orange ( $\\textbf{O}$ ), and purple ( $\\textbf{P}$ ). Call an arrangement of blocks $\\textit{even}$ if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement \\[\\textbf{R B B Y G G Y R O P P O}\\] is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is $\\frac{m}{n},$ 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\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 \\[\\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.\\]",
"type": "text"
}
],
"role": "user"
}
] |
$373$
|
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": "Compute \\[\\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.\\]",
"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": "In a parlor game, the magician asks one of the participants to think of a three digit number $(abc)$ where $a$ , $b$ , and $c$ represent digits in base $10$ in the order indicated. The magician then asks this person to form the numbers $(acb)$ , $(bca)$ , $(bac)$ , $(cab)$ , and $(cba)$ , to add these five numbers, and to reveal their sum, $N$ . If told the value of $N$ , the magician can identify the original number, $(abc)$ . Play the role of the magician and determine the $(abc)$ if $N= 3194$ .",
"type": "text"
}
],
"role": "user"
}
] |
$358$
|
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": "In a parlor game, the magician asks one of the participants to think of a three digit number $(abc)$ where $a$ , $b$ , and $c$ represent digits in base $10$ in the order indicated. The magician then asks this person to form the numbers $(acb)$ , $(bca)$ , $(bac)$ , $(cab)$ , and $(cba)$ , to add these five numbers, and to reveal their sum, $N$ . If told the value of $N$ , the magician can identify the original number, $(abc)$ . Play the role of the magician and determine the $(abc)$ if $N= 3194$ .",
"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{}:\nRecall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292 = 444_{\\text{eight}}.$",
"type": "text"
}
],
"role": "user"
}
] |
$585$
|
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{}:\nRecall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292 = 444_{\\text{eight}}.$",
"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 drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\\frac{1}{2}$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?",
"type": "text"
}
],
"role": "user"
}
] |
$990$
|
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 drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\\frac{1}{2}$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?",
"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{}:\nSegments $\\overline{AB}, \\overline{AC},$ and $\\overline{AD}$ are edges of a cube and $\\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\\sqrt{10}$ , $CP=60\\sqrt{5}$ , $DP=120\\sqrt{2}$ , and $GP=36\\sqrt{7}$ . Find $AP.$",
"type": "text"
}
],
"role": "user"
}
] |
$192$
|
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{}:\nSegments $\\overline{AB}, \\overline{AC},$ and $\\overline{AD}$ are edges of a cube and $\\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\\sqrt{10}$ , $CP=60\\sqrt{5}$ , $DP=120\\sqrt{2}$ , and $GP=36\\sqrt{7}$ . Find $AP.$",
"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 permutation $p = (a_1,a_2,\\ldots,a_9)$ of the digits $1,2,\\ldots,9$ , let $s(p)$ denote the sum of the three $3$ -digit numbers $a_1a_2a_3$ , $a_4a_5a_6$ , and $a_7a_8a_9$ . Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$ . Let $n$ denote the number of permutations $p$ with $s(p) = m$ . Find $|m - n|$ .",
"type": "text"
}
],
"role": "user"
}
] |
$162$
|
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 permutation $p = (a_1,a_2,\\ldots,a_9)$ of the digits $1,2,\\ldots,9$ , let $s(p)$ denote the sum of the three $3$ -digit numbers $a_1a_2a_3$ , $a_4a_5a_6$ , and $a_7a_8a_9$ . Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$ . Let $n$ denote the number of permutations $p$ with $s(p) = m$ . 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 biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that 25% of these fish are no longer in the lake on September 1 (because of death and emigrations), that 40% of the fish were not in the lake May 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the September 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May 1?",
"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",
"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 biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that 25% of these fish are no longer in the lake on September 1 (because of death and emigrations), that 40% of the fish were not in the lake May 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the September 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May 1?",
"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{}:\nThere is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed? [asy] size(50); pair A,B; A=(0,0); B=(2,0); pair C=rotate(60,A)*B; pair D, E, F; D = (1,0); E=rotate(60,A)*D; F=rotate(60,C)*E; draw(C--A--B--cycle); draw(D--E--F--cycle); [/asy]",
"type": "text"
}
],
"role": "user"
}
] |
$336$
|
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{}:\nThere is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed? [asy] size(50); pair A,B; A=(0,0); B=(2,0); pair C=rotate(60,A)*B; pair D, E, F; D = (1,0); E=rotate(60,A)*D; F=rotate(60,C)*E; draw(C--A--B--cycle); draw(D--E--F--cycle); [/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{}:\nSuppose that the roots of $x^3+3x^2+4x-11=0$ are $a$ , $b$ , and $c$ , and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$ , $b+c$ , and $c+a$ . Find $t$ .",
"type": "text"
}
],
"role": "user"
}
] |
$23$
|
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{}:\nSuppose that the roots of $x^3+3x^2+4x-11=0$ are $a$ , $b$ , and $c$ , and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$ , $b+c$ , and $c+a$ . Find $t$ .",
"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 least positive integer such that when its leftmost digit is deleted, the resulting integer is $1/29$ of the original integer.",
"type": "text"
}
],
"role": "user"
}
] |
$725$
|
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 least positive integer such that when its leftmost digit is deleted, the resulting integer is $1/29$ of the original 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",
"type": "text"
}
],
"role": "system"
},
{
"content": [
{
"text": "You will be given a problem. Please reason step by step, and put your final answer within \\boxed{}:\nCircles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2.$ Given that the length of the chord is $\\frac{m\\sqrt{n}}p$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p.$",
"type": "text"
}
],
"role": "user"
}
] |
$405$
|
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{}:\nCircles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2.$ Given that the length of the chord is $\\frac{m\\sqrt{n}}p$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ 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{}:\nFind the number of positive integers with three not necessarily distinct digits, $abc$ , with $a \\neq 0$ and $c \\neq 0$ such that both $abc$ and $cba$ are multiples of $4$ .",
"type": "text"
}
],
"role": "user"
}
] |
$40$
|
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 number of positive integers with three not necessarily distinct digits, $abc$ , with $a \\neq 0$ and $c \\neq 0$ such that both $abc$ and $cba$ are multiples of $4$ .",
"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": "Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point $D$ on the south edge of the field. Cao arrives at point $D$ at the same time that Ana and Bob arrive at $D$ for the first time. The ratio of the field's length to the field's width to the distance from point $D$ to the southeast corner of the field can be represented as $p : q : r$ , where $p$ , $q$ , and $r$ are positive integers with $p$ and $q$ relatively prime. Find $p+q+r$ .",
"type": "text"
}
],
"role": "user"
}
] |
$61$
|
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": "Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point $D$ on the south edge of the field. Cao arrives at point $D$ at the same time that Ana and Bob arrive at $D$ for the first time. The ratio of the field's length to the field's width to the distance from point $D$ to the southeast corner of the field can be represented as $p : q : r$ , where $p$ , $q$ , and $r$ are positive integers with $p$ and $q$ relatively prime. Find $p+q+r$ .",
"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 $\\triangle{PQR}$ be a right triangle with $PQ = 90$ , $PR = 120$ , and $QR = 150$ . Let $C_{1}$ be the inscribed circle. Construct $\\overline{ST}$ with $S$ on $\\overline{PR}$ and $T$ on $\\overline{QR}$ , such that $\\overline{ST}$ is perpendicular to $\\overline{PR}$ and tangent to $C_{1}$ . Construct $\\overline{UV}$ with $U$ on $\\overline{PQ}$ and $V$ on $\\overline{QR}$ such that $\\overline{UV}$ is perpendicular to $\\overline{PQ}$ and tangent to $C_{1}$ . Let $C_{2}$ be the inscribed circle of $\\triangle{RST}$ and $C_{3}$ the inscribed circle of $\\triangle{QUV}$ . The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\\sqrt {10n}$ . What is $n$ ?",
"type": "text"
}
],
"role": "user"
}
] |
$725$
|
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 $\\triangle{PQR}$ be a right triangle with $PQ = 90$ , $PR = 120$ , and $QR = 150$ . Let $C_{1}$ be the inscribed circle. Construct $\\overline{ST}$ with $S$ on $\\overline{PR}$ and $T$ on $\\overline{QR}$ , such that $\\overline{ST}$ is perpendicular to $\\overline{PR}$ and tangent to $C_{1}$ . Construct $\\overline{UV}$ with $U$ on $\\overline{PQ}$ and $V$ on $\\overline{QR}$ such that $\\overline{UV}$ is perpendicular to $\\overline{PQ}$ and tangent to $C_{1}$ . Let $C_{2}$ be the inscribed circle of $\\triangle{RST}$ and $C_{3}$ the inscribed circle of $\\triangle{QUV}$ . The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\\sqrt {10n}$ . What is $n$ ?",
"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{}:\nSixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of the 16 chairs that could be selected. Find the remainder when $N$ is divided by $1000$.",
"type": "text"
}
],
"role": "user"
}
] |
$907$
|
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",
"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{}:\nSixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of the 16 chairs that could be selected. 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": "The sequence $(a_n)$ satisfies $a_0=0$ and $a_{n + 1} = \\frac{8}{5}a_n + \\frac{6}{5}\\sqrt{4^n - a_n^2}$ for $n \\geq 0$ . Find the greatest integer less than or equal to $a_{10}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$983$
|
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 $(a_n)$ satisfies $a_0=0$ and $a_{n + 1} = \\frac{8}{5}a_n + \\frac{6}{5}\\sqrt{4^n - a_n^2}$ for $n \\geq 0$ . Find the greatest integer less than or equal to $a_{10}$ .",
"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{}:\nEach of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M-m$ .",
"type": "text"
}
],
"role": "user"
}
] |
$298$
|
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{}:\nEach of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M-m$ .",
"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": "Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface . The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$ . It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its circumference as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a distance $x$ from where it starts. The distance $x$ can be expressed in the form $a\\pi+b\\sqrt{c},$ where $a,$ $b,$ and $c$ are integers and $c$ is not divisible by the square of any prime . Find $a+b+c.$",
"type": "text"
}
],
"role": "user"
}
] |
$179$
|
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": "Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface . The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$ . It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its circumference as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a distance $x$ from where it starts. The distance $x$ can be expressed in the form $a\\pi+b\\sqrt{c},$ where $a,$ $b,$ and $c$ are integers and $c$ is not divisible by the square of any prime . 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",
"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 least positive integer $k$ for which the equation $\\left\\lfloor\\frac{2002}{n}\\right\\rfloor=k$ has no integer solutions for $n$ . (The notation $\\lfloor x\\rfloor$ means the greatest integer less than or equal to $x$ .)",
"type": "text"
}
],
"role": "user"
}
] |
$49$
|
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 least positive integer $k$ for which the equation $\\left\\lfloor\\frac{2002}{n}\\right\\rfloor=k$ has no integer solutions for $n$ . (The notation $\\lfloor x\\rfloor$ means the greatest integer 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{}:\nThree concentric circles have radii $3,$ $4,$ and $5.$ An equilateral triangle with one vertex on each circle has side length $s.$ The largest possible area of the triangle can be written as $a + \\tfrac{b}{c} \\sqrt{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not divisible by the square of any prime. Find $a+b+c+d.$",
"type": "text"
}
],
"role": "user"
}
] |
$41$
|
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{}:\nThree concentric circles have radii $3,$ $4,$ and $5.$ An equilateral triangle with one vertex on each circle has side length $s.$ The largest possible area of the triangle can be written as $a + \\tfrac{b}{c} \\sqrt{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ 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": "Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ ",
"type": "text"
}
],
"role": "user"
}
] |
$191$
|
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": "Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\\frac{m}{n},$ 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{}:\nIn a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when $N$ is divided by $1000$ .",
"type": "text"
}
],
"role": "user"
}
] |
$16$
|
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 a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. 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{}:\nLet $A_1 A_2 A_3 \\ldots A_{11}$ be an $11$-sided non-convex simple polygon with the following properties:\n\n\\begin{itemize}\n\\item For every integer $2 \\le i \\le 10$, the area of $\\triangle A_i A_{1} A_{i+1}$ is equal to $1$.\n\\item For every integer $2 \\le i \\le 10$, $\\cos(\\angle A_i A_{1} A_{i+1}) = \\frac{12}{13}$.\n\\item The perimeter of the $11$-gon $A_1 A_2 A_3 \\dots A_{11}$ is equal to $20$.\n\\end{itemize}\n\nThen $A_1 A_2 + A_1 A_{11} = \\frac{m \\sqrt{n} - p}{q}$ where $m, n, p$, and $q$ are positive integers, $n$ is not divisible by the square of any prime, and no prime divides all of $m, p$, and $q$. Find $m + n + p + q$.",
"type": "text"
}
],
"role": "user"
}
] |
$19$
|
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",
"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 $A_1 A_2 A_3 \\ldots A_{11}$ be an $11$-sided non-convex simple polygon with the following properties:\n\n\\begin{itemize}\n\\item For every integer $2 \\le i \\le 10$, the area of $\\triangle A_i A_{1} A_{i+1}$ is equal to $1$.\n\\item For every integer $2 \\le i \\le 10$, $\\cos(\\angle A_i A_{1} A_{i+1}) = \\frac{12}{13}$.\n\\item The perimeter of the $11$-gon $A_1 A_2 A_3 \\dots A_{11}$ is equal to $20$.\n\\end{itemize}\n\nThen $A_1 A_2 + A_1 A_{11} = \\frac{m \\sqrt{n} - p}{q}$ where $m, n, p$, and $q$ are positive integers, $n$ is not divisible by the square of any prime, and no prime divides all of $m, p$, and $q$. 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",
"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 hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\\overline{AB}$ , has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$384$
|
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 hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\\overline{AB}$ , has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$ .",
"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": "Triangle $ABC$ has $AC = 450$ and $BC = 300$ . Points $K$ and $L$ are located on $\\overline{AC}$ and $\\overline{AB}$ respectively so that $AK = CK$ , and $\\overline{CL}$ is the angle bisector of angle $C$ . Let $P$ be the point of intersection of $\\overline{BK}$ and $\\overline{CL}$ , and let $M$ be the point on line $BK$ for which $K$ is the midpoint of $\\overline{PM}$ . If $AM = 180$ , find $LP$ .",
"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\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": "Triangle $ABC$ has $AC = 450$ and $BC = 300$ . Points $K$ and $L$ are located on $\\overline{AC}$ and $\\overline{AB}$ respectively so that $AK = CK$ , and $\\overline{CL}$ is the angle bisector of angle $C$ . Let $P$ be the point of intersection of $\\overline{BK}$ and $\\overline{CL}$ , and let $M$ be the point on line $BK$ for which $K$ is the midpoint of $\\overline{PM}$ . If $AM = 180$ , find $LP$ .",
"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{}:\nJenn randomly chooses a number $J$ from $1, 2, 3,\\ldots, 19, 20$ . Bela then randomly chooses a number $B$ from $1, 2, 3,\\ldots, 19, 20$ distinct from $J$ . The value of $B - J$ is at least $2$ with a probability that 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"
}
] |
$29$
|
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{}:\nJenn randomly chooses a number $J$ from $1, 2, 3,\\ldots, 19, 20$ . Bela then randomly chooses a number $B$ from $1, 2, 3,\\ldots, 19, 20$ distinct from $J$ . The value of $B - J$ is at least $2$ with a probability that 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\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": "Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\\frac {1}{3}$ . The probability that Club Truncator will finish the season with more wins than losses is $\\frac {m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$341$
|
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": "Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\\frac {1}{3}$ . The probability that Club Truncator will finish the season with more wins than losses is $\\frac {m}{n}$ , 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{}:\nIn parallelogram $ABCD$ , point $M$ is on $\\overline{AB}$ so that $\\frac {AM}{AB} = \\frac {17}{1000}$ and point $N$ is on $\\overline{AD}$ so that $\\frac {AN}{AD} = \\frac {17}{2009}$ . Let $P$ be the point of intersection of $\\overline{AC}$ and $\\overline{MN}$ . Find $\\frac {AC}{AP}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$177$
|
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 parallelogram $ABCD$ , point $M$ is on $\\overline{AB}$ so that $\\frac {AM}{AB} = \\frac {17}{1000}$ and point $N$ is on $\\overline{AD}$ so that $\\frac {AN}{AD} = \\frac {17}{2009}$ . Let $P$ be the point of intersection of $\\overline{AC}$ and $\\overline{MN}$ . Find $\\frac {AC}{AP}$ .",
"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{}:\nThree spheres with radii $11,$ $13,$ and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A,$ $B,$ and $C,$ respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that $AB^2 = 560.$ Find $AC^2.$",
"type": "text"
}
],
"role": "user"
}
] |
$756$
|
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{}:\nThree spheres with radii $11,$ $13,$ and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A,$ $B,$ and $C,$ respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that $AB^2 = 560.$ Find $AC^2.$",
"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 inscribed circle of triangle $ABC$ is tangent to $\\overline{AB}$ at $P_{},$ and its radius is 21. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle.",
"type": "text"
}
],
"role": "user"
}
] |
$345$
|
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 inscribed circle of triangle $ABC$ is tangent to $\\overline{AB}$ at $P_{},$ and its radius is 21. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle.",
"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": "Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \\%,$ and $m = a - b\\sqrt {c},$ where $a, b,$ and $c$ are positive integers , and $c$ is not divisible by the square of any prime . Find $a + b + c.$",
"type": "text"
}
],
"role": "user"
}
] |
$87$
|
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": "Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \\%,$ and $m = a - b\\sqrt {c},$ where $a, b,$ and $c$ are positive integers , and $c$ is not divisible by the square of any prime . 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",
"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 each positive integer $n$ , let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$ . For example, $f(2020) = f(133210_{\\text{4}}) = 10 = 12_{\\text{8}}$ , and $g(2020) = \\text{the digit sum of }12_{\\text{8}} = 3$ . Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9$ . Find the remainder when $N$ is divided by $1000$ .",
"type": "text"
}
],
"role": "user"
}
] |
$151$
|
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 each positive integer $n$ , let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$ . For example, $f(2020) = f(133210_{\\text{4}}) = 10 = 12_{\\text{8}}$ , and $g(2020) = \\text{the digit sum of }12_{\\text{8}} = 3$ . Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9$ . 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": "Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$ , and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\\triangle GEM$ .",
"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\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 $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$ , and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\\triangle GEM$ .",
"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 $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$ , and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$ . Find the remainder when $m-n$ is divided by $1000$ .",
"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{}:\nLet $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$ , and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$ . 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": "Let $\\tau(n)$ denote the number of positive integer divisors of $n$ . Find the sum of the six least positive integers $n$ that are solutions to $\\tau (n) + \\tau (n+1) = 7$ .",
"type": "text"
}
],
"role": "user"
}
] |
$540$
|
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 $\\tau(n)$ denote the number of positive integer divisors of $n$ . Find the sum of the six least positive integers $n$ that are solutions to $\\tau (n) + \\tau (n+1) = 7$ .",
"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": "Set $A$ consists of $m$ consecutive integers whose sum is $2m,$ and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is 99. Find $m.$",
"type": "text"
}
],
"role": "user"
}
] |
$201$
|
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": "Set $A$ consists of $m$ consecutive integers whose sum is $2m,$ and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is 99. Find $m.$",
"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": "Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator? ",
"type": "text"
}
],
"role": "user"
}
] |
$57$
|
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": "Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator? ",
"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 flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7$ . Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .",
"type": "text"
}
],
"role": "user"
}
] |
$173$
|
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 flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7$ . Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\\tfrac{m}{n}$ , 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{}:\nFor each integer $n \\ge 2$ , let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\\le x \\le n$ and $0\\le y \\le x \\left\\lfloor \\sqrt x \\right\\rfloor$ , where $\\left\\lfloor \\sqrt x \\right\\rfloor$ is the greatest integer not exceeding $\\sqrt x$ . Find the number of values of $n$ with $2\\le n \\le 1000$ for which $A(n)$ is an integer.",
"type": "text"
}
],
"role": "user"
}
] |
$483$
|
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 each integer $n \\ge 2$ , let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\\le x \\le n$ and $0\\le y \\le x \\left\\lfloor \\sqrt x \\right\\rfloor$ , where $\\left\\lfloor \\sqrt x \\right\\rfloor$ is the greatest integer not exceeding $\\sqrt x$ . Find the number of values of $n$ with $2\\le n \\le 1000$ for which $A(n)$ 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 set $\\mathcal{A}$ be a 90-element subset of $\\{1,2,3,\\ldots,100\\},$ and let $S$ be the sum of the elements of $\\mathcal{A}.$ Find the number of possible values of $S.$",
"type": "text"
}
],
"role": "user"
}
] |
$901$
|
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 set $\\mathcal{A}$ be a 90-element subset of $\\{1,2,3,\\ldots,100\\},$ and let $S$ be the sum of the elements of $\\mathcal{A}.$ Find the number of possible values of $S.$",
"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 jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$",
"type": "text"
}
],
"role": "user"
}
] |
$441$
|
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 jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, 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{}:\nFor any sequence of real numbers $A=(a_1,a_2,a_3,\\ldots)$ , define $\\Delta A^{}_{}$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\\ldots)$ , whose $n^{th}$ term is $a_{n+1}-a_n^{}$ . Suppose that all of the terms of the sequence $\\Delta(\\Delta A^{}_{})$ are $1^{}_{}$ , and that $a_{19}=a_{92}^{}=0$ . Find $a_1^{}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$819$
|
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 any sequence of real numbers $A=(a_1,a_2,a_3,\\ldots)$ , define $\\Delta A^{}_{}$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\\ldots)$ , whose $n^{th}$ term is $a_{n+1}-a_n^{}$ . Suppose that all of the terms of the sequence $\\Delta(\\Delta A^{}_{})$ are $1^{}_{}$ , and that $a_{19}=a_{92}^{}=0$ . Find $a_1^{}$ .",
"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 fair coin is to be tossed $10_{}^{}$ times. Let $i/j^{}_{}$ , in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j_{}^{}$ .",
"type": "text"
}
],
"role": "user"
}
] |
$73$
|
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 fair coin is to be tossed $10_{}^{}$ times. Let $i/j^{}_{}$ , in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j_{}^{}$ .",
"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 number of subsets of $\\{1,2,3,\\ldots,10\\}$ that contain exactly one pair of consecutive integers. Examples of such subsets are $\\{\\mathbf{1},\\mathbf{2},5\\}$ and $\\{1,3,\\mathbf{6},\\mathbf{7},10\\}.$",
"type": "text"
}
],
"role": "user"
}
] |
$235$
|
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 number of subsets of $\\{1,2,3,\\ldots,10\\}$ that contain exactly one pair of consecutive integers. Examples of such subsets are $\\{\\mathbf{1},\\mathbf{2},5\\}$ and $\\{1,3,\\mathbf{6},\\mathbf{7},10\\}.$",
"type": "text",
"url": null
}
],
"role": "user"
}
] |
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