data_source stringclasses 1
value | ability stringclasses 1
value | reward_model dict | prompt listlengths 2 2 |
|---|---|---|---|
aime_26 | mathematics | {
"ground_truth": "277",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Patrick started walking at a constant rate along a straight road from school to the park. One hour after Patrick left, Tanya started running along the same road from school to th... |
aime_26 | mathematics | {
"ground_truth": "62",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of positive integer palindromes written in base $10$ with no zero digits, and whose digits add up to $13$. For example, $42124$ has these properties. Recall that ... |
aime_26 | mathematics | {
"ground_truth": "79",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A hemisphere with radius $200$ sits on top of a horizontal circular disk with radius $200,$ and the hemisphere and disk have the same center. Let $\\mathcal T$ be the region of p... |
aime_26 | mathematics | {
"ground_truth": "70",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of integers less than or equal to 100 that are equal to $a+b+ab$ for some choice of distinct positive integers a and b.",
"role": "user"
}
] |
aime_26 | mathematics | {
"ground_truth": "65",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\\theta$ around point $B$ to point $A^\\prime$. Then... |
aime_26 | mathematics | {
"ground_truth": "441",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A real number $x$ satisfies $\\sqrt[20]{x^{\\log_{2026}x}}=26x$. What is the number of positive divisors of the product of all possible positive values of $x$?",
"role": "use... |
aime_26 | mathematics | {
"ground_truth": "396",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "\nFind the number of functions $\\pi$ mapping the set $A =\\{1,2,3,4,5,6\\}$ onto $A$ such that for every $a \\in A,$\n\\[\n\\pi(\\pi(\\pi(\\pi(\\pi(\\pi(a)))))) = a.\n\\]",
... |
aime_26 | mathematics | {
"ground_truth": "244",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $N$ be the number of positive integer divisors of $17017^{17}$ that leave a remainder of $5$ when divided by $12$. Find the remainder when $N$ is divided by $1000$.",
"ro... |
aime_26 | mathematics | {
"ground_truth": "29",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Joanne has a blank fair six-sided die and six stickers each displaying a different integer from 1 to 6. Joanne rolls the die and then places the sticker labeled 1 on the top face... |
aime_26 | mathematics | {
"ground_truth": "156",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ have side lengths $AB = 13, BC = 14,$ and $CA = 15.$ Triangle $\\triangle A'B'C'$ is obtained by rotating $\\triangle ABC$ about its circumcenter so that ${}... |
aime_26 | mathematics | {
"ground_truth": "896",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The integers from $1$ to $64$ are placed in some order into an $8 \\times 8$ grid of cells with one number in each cell. Let $a_{i,j}$ be the number placed in the cell in row $i$... |
aime_26 | mathematics | {
"ground_truth": "161",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Triangle $\\triangle ABC$ lies in plane $\\mathcal P$ with $AB = 6, AC = 4,$ and $\\angle BAC = 90^\\circ.$ Let $D$ be the reflection across $\\overline{BC}$ of the centroid of $... |
aime_26 | mathematics | {
"ground_truth": "39",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each positive integer $r$ less than $502,$ define\n\\[\nS_r=\\sum_{m\\ge 0}\\dbinom{10000}{502n+r},\n\\]\nwhere $\\binom{10000}{n}$ is defined to be $0$ when $n>10000.$ That ... |
aime_26 | mathematics | {
"ground_truth": "681",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In an equiangular pentagon, the sum of the squares of the side lengths equals $308,$ and the sum of the squares of the diagonal lengths equals $800.$ The square of the perimeter ... |
aime_26 | mathematics | {
"ground_truth": "83",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a, b,$ and $n$ be positive integers with both $a$ and $b$ greater than or equal to $2$ and less than or equal to $2n$. Define an $a \\times b$ cell loop in a $2n \\times 2n$... |
aime_26 | mathematics | {
"ground_truth": "178",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of the $10$th terms of all arithmetic sequences of integers that have first term equal to $4$ and include both $24$ and $34$ as terms.",
"role": "user"
}
] |
aime_26 | mathematics | {
"ground_truth": "243",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The figure below shows a grid of $10$ squares in a row. Each square has a diagonal connecting its lower left vertex to its upper right vertex. A bug moves along the line segments... |
aime_26 | mathematics | {
"ground_truth": "503",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABCDE$ be a nonconvex pentagon with internal angles $\\angle A = \\angle E = 90^\\circ$ and $\\angle B = \\angle D = 45^\\circ.$ Suppose that $DE < AB, AE = 20, BC = 14\\sqr... |
aime_26 | mathematics | {
"ground_truth": "279",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each positive integer $n$ let $f(n)$ be the value of the base-ten numeral $n$ viewed in base $b$, where $b$ is the least integer greater than the greatest digit in $n$. For e... |
aime_26 | mathematics | {
"ground_truth": "190",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An urn contains $n$ marbles. Each marble is either red or blue, and there are at least $7$ marbles of each color. When $7$ marbles are drawn randomly from the urn without replace... |
aime_26 | mathematics | {
"ground_truth": "50",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of all real numbers $r$ such that there is at least one point where the circle with radius $r$ centered at $(4, 39)$ is tangent to the parabola with equation $2y = x... |
aime_26 | mathematics | {
"ground_truth": "754",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A standard fair six-sided die is rolled repeatedly. Each time the die reads 1 or 2, Alice gets a coin; each time it reads 3 or 4, Bob gets a coin; and each time it reads 5 or 6, ... |
aime_26 | mathematics | {
"ground_truth": "245",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Isosceles triangle $\\triangle ABC$ has $AB = BC.$ Let $I$ be the incenter of $\\triangle ABC.$ The perimeters of $\\triangle ABC$ and $\\triangle AIC$ are in the ratio $125:6,$ ... |
aime_26 | mathematics | {
"ground_truth": "669",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ denote the value of the infinite sum\n\\[\n\\frac{1}{9} + \\frac{1}{99} + \\frac{1}{999} + \\frac{1}{9999} + \\cdots\n\\]\nFind the remainder when the greatest integer l... |
aime_26 | mathematics | {
"ground_truth": "850",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ be a triangle with $D$ on $\\overline{BC}$ such that $\\overline{AD}$ bisects $\\angle BAC.$ Let $\\omega$ be the circle that passes through $A$ and is tange... |
aime_26 | mathematics | {
"ground_truth": "132",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the greatest integer $n$ such that the cubic polynomial\n\\[\nx^{3} - \\displaystyle\\frac{n}{6}x^{2} + (n - 11)x - 400\n\\]\nhas roots $\\alpha^{2}$, $\\beta^{2}$, and $\\g... |
aime_26 | mathematics | {
"ground_truth": "223",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a tetrahedron with two isosceles triangle faces with side lengths $5\\sqrt{10}, 5\\sqrt{10},$ and $10$ and two isosceles triangle faces with side lengths $5\\sqrt{10}, 5... |
aime_26 | mathematics | {
"ground_truth": "107",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Call finite sets of integers $S$ and $T$ cousins if\n- $S$ and $T$ have the same number of elements,\n- $S$ and $T$ are disjoint, and\n- the elements of $S$ can be paired with th... |
aime_26 | mathematics | {
"ground_truth": "157",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For integers $a$ and $b,$ let $a \\circ b = a - b$ if $a$ is odd and $b$ is even, and $a+b$ otherwise. Find the number of sequences $a_1, a_2, a_3, \\ldots, a_n$ of positive inte... |
aime_26 | mathematics | {
"ground_truth": "393",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of ordered 7-tuples $(a_1, a_2, a_3, \\ldots, a_7)$ having the following properties:\n- $a_k \\in \\{1,2,3\\}$ for all $k.$\n- $a_1+a_2+a_3+a_4+a_5+a_6+a_7$ is a ... |
aime_26 | mathematics | {
"ground_truth": "277",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Patrick started walking at a constant rate along a straight road from school to the park. One hour after Patrick left, Tanya started running along the same road from school to th... |
aime_26 | mathematics | {
"ground_truth": "62",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of positive integer palindromes written in base $10$ with no zero digits, and whose digits add up to $13$. For example, $42124$ has these properties. Recall that ... |
aime_26 | mathematics | {
"ground_truth": "79",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A hemisphere with radius $200$ sits on top of a horizontal circular disk with radius $200,$ and the hemisphere and disk have the same center. Let $\\mathcal T$ be the region of p... |
aime_26 | mathematics | {
"ground_truth": "70",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of integers less than or equal to 100 that are equal to $a+b+ab$ for some choice of distinct positive integers a and b.",
"role": "user"
}
] |
aime_26 | mathematics | {
"ground_truth": "65",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\\theta$ around point $B$ to point $A^\\prime$. Then... |
aime_26 | mathematics | {
"ground_truth": "441",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A real number $x$ satisfies $\\sqrt[20]{x^{\\log_{2026}x}}=26x$. What is the number of positive divisors of the product of all possible positive values of $x$?",
"role": "use... |
aime_26 | mathematics | {
"ground_truth": "396",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "\nFind the number of functions $\\pi$ mapping the set $A =\\{1,2,3,4,5,6\\}$ onto $A$ such that for every $a \\in A,$\n\\[\n\\pi(\\pi(\\pi(\\pi(\\pi(\\pi(a)))))) = a.\n\\]",
... |
aime_26 | mathematics | {
"ground_truth": "244",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $N$ be the number of positive integer divisors of $17017^{17}$ that leave a remainder of $5$ when divided by $12$. Find the remainder when $N$ is divided by $1000$.",
"ro... |
aime_26 | mathematics | {
"ground_truth": "29",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Joanne has a blank fair six-sided die and six stickers each displaying a different integer from 1 to 6. Joanne rolls the die and then places the sticker labeled 1 on the top face... |
aime_26 | mathematics | {
"ground_truth": "156",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ have side lengths $AB = 13, BC = 14,$ and $CA = 15.$ Triangle $\\triangle A'B'C'$ is obtained by rotating $\\triangle ABC$ about its circumcenter so that ${}... |
aime_26 | mathematics | {
"ground_truth": "896",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The integers from $1$ to $64$ are placed in some order into an $8 \\times 8$ grid of cells with one number in each cell. Let $a_{i,j}$ be the number placed in the cell in row $i$... |
aime_26 | mathematics | {
"ground_truth": "161",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Triangle $\\triangle ABC$ lies in plane $\\mathcal P$ with $AB = 6, AC = 4,$ and $\\angle BAC = 90^\\circ.$ Let $D$ be the reflection across $\\overline{BC}$ of the centroid of $... |
aime_26 | mathematics | {
"ground_truth": "39",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each positive integer $r$ less than $502,$ define\n\\[\nS_r=\\sum_{m\\ge 0}\\dbinom{10000}{502n+r},\n\\]\nwhere $\\binom{10000}{n}$ is defined to be $0$ when $n>10000.$ That ... |
aime_26 | mathematics | {
"ground_truth": "681",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In an equiangular pentagon, the sum of the squares of the side lengths equals $308,$ and the sum of the squares of the diagonal lengths equals $800.$ The square of the perimeter ... |
aime_26 | mathematics | {
"ground_truth": "83",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a, b,$ and $n$ be positive integers with both $a$ and $b$ greater than or equal to $2$ and less than or equal to $2n$. Define an $a \\times b$ cell loop in a $2n \\times 2n$... |
aime_26 | mathematics | {
"ground_truth": "178",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of the $10$th terms of all arithmetic sequences of integers that have first term equal to $4$ and include both $24$ and $34$ as terms.",
"role": "user"
}
] |
aime_26 | mathematics | {
"ground_truth": "243",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The figure below shows a grid of $10$ squares in a row. Each square has a diagonal connecting its lower left vertex to its upper right vertex. A bug moves along the line segments... |
aime_26 | mathematics | {
"ground_truth": "503",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABCDE$ be a nonconvex pentagon with internal angles $\\angle A = \\angle E = 90^\\circ$ and $\\angle B = \\angle D = 45^\\circ.$ Suppose that $DE < AB, AE = 20, BC = 14\\sqr... |
aime_26 | mathematics | {
"ground_truth": "279",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each positive integer $n$ let $f(n)$ be the value of the base-ten numeral $n$ viewed in base $b$, where $b$ is the least integer greater than the greatest digit in $n$. For e... |
aime_26 | mathematics | {
"ground_truth": "190",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An urn contains $n$ marbles. Each marble is either red or blue, and there are at least $7$ marbles of each color. When $7$ marbles are drawn randomly from the urn without replace... |
aime_26 | mathematics | {
"ground_truth": "50",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of all real numbers $r$ such that there is at least one point where the circle with radius $r$ centered at $(4, 39)$ is tangent to the parabola with equation $2y = x... |
aime_26 | mathematics | {
"ground_truth": "754",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A standard fair six-sided die is rolled repeatedly. Each time the die reads 1 or 2, Alice gets a coin; each time it reads 3 or 4, Bob gets a coin; and each time it reads 5 or 6, ... |
aime_26 | mathematics | {
"ground_truth": "245",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Isosceles triangle $\\triangle ABC$ has $AB = BC.$ Let $I$ be the incenter of $\\triangle ABC.$ The perimeters of $\\triangle ABC$ and $\\triangle AIC$ are in the ratio $125:6,$ ... |
aime_26 | mathematics | {
"ground_truth": "669",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ denote the value of the infinite sum\n\\[\n\\frac{1}{9} + \\frac{1}{99} + \\frac{1}{999} + \\frac{1}{9999} + \\cdots\n\\]\nFind the remainder when the greatest integer l... |
aime_26 | mathematics | {
"ground_truth": "850",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ be a triangle with $D$ on $\\overline{BC}$ such that $\\overline{AD}$ bisects $\\angle BAC.$ Let $\\omega$ be the circle that passes through $A$ and is tange... |
aime_26 | mathematics | {
"ground_truth": "132",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the greatest integer $n$ such that the cubic polynomial\n\\[\nx^{3} - \\displaystyle\\frac{n}{6}x^{2} + (n - 11)x - 400\n\\]\nhas roots $\\alpha^{2}$, $\\beta^{2}$, and $\\g... |
aime_26 | mathematics | {
"ground_truth": "223",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a tetrahedron with two isosceles triangle faces with side lengths $5\\sqrt{10}, 5\\sqrt{10},$ and $10$ and two isosceles triangle faces with side lengths $5\\sqrt{10}, 5... |
aime_26 | mathematics | {
"ground_truth": "107",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Call finite sets of integers $S$ and $T$ cousins if\n- $S$ and $T$ have the same number of elements,\n- $S$ and $T$ are disjoint, and\n- the elements of $S$ can be paired with th... |
aime_26 | mathematics | {
"ground_truth": "157",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For integers $a$ and $b,$ let $a \\circ b = a - b$ if $a$ is odd and $b$ is even, and $a+b$ otherwise. Find the number of sequences $a_1, a_2, a_3, \\ldots, a_n$ of positive inte... |
aime_26 | mathematics | {
"ground_truth": "393",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of ordered 7-tuples $(a_1, a_2, a_3, \\ldots, a_7)$ having the following properties:\n- $a_k \\in \\{1,2,3\\}$ for all $k.$\n- $a_1+a_2+a_3+a_4+a_5+a_6+a_7$ is a ... |
aime_26 | mathematics | {
"ground_truth": "277",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Patrick started walking at a constant rate along a straight road from school to the park. One hour after Patrick left, Tanya started running along the same road from school to th... |
aime_26 | mathematics | {
"ground_truth": "62",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of positive integer palindromes written in base $10$ with no zero digits, and whose digits add up to $13$. For example, $42124$ has these properties. Recall that ... |
aime_26 | mathematics | {
"ground_truth": "79",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A hemisphere with radius $200$ sits on top of a horizontal circular disk with radius $200,$ and the hemisphere and disk have the same center. Let $\\mathcal T$ be the region of p... |
aime_26 | mathematics | {
"ground_truth": "70",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of integers less than or equal to 100 that are equal to $a+b+ab$ for some choice of distinct positive integers a and b.",
"role": "user"
}
] |
aime_26 | mathematics | {
"ground_truth": "65",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\\theta$ around point $B$ to point $A^\\prime$. Then... |
aime_26 | mathematics | {
"ground_truth": "441",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A real number $x$ satisfies $\\sqrt[20]{x^{\\log_{2026}x}}=26x$. What is the number of positive divisors of the product of all possible positive values of $x$?",
"role": "use... |
aime_26 | mathematics | {
"ground_truth": "396",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "\nFind the number of functions $\\pi$ mapping the set $A =\\{1,2,3,4,5,6\\}$ onto $A$ such that for every $a \\in A,$\n\\[\n\\pi(\\pi(\\pi(\\pi(\\pi(\\pi(a)))))) = a.\n\\]",
... |
aime_26 | mathematics | {
"ground_truth": "244",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $N$ be the number of positive integer divisors of $17017^{17}$ that leave a remainder of $5$ when divided by $12$. Find the remainder when $N$ is divided by $1000$.",
"ro... |
aime_26 | mathematics | {
"ground_truth": "29",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Joanne has a blank fair six-sided die and six stickers each displaying a different integer from 1 to 6. Joanne rolls the die and then places the sticker labeled 1 on the top face... |
aime_26 | mathematics | {
"ground_truth": "156",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ have side lengths $AB = 13, BC = 14,$ and $CA = 15.$ Triangle $\\triangle A'B'C'$ is obtained by rotating $\\triangle ABC$ about its circumcenter so that ${}... |
aime_26 | mathematics | {
"ground_truth": "896",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The integers from $1$ to $64$ are placed in some order into an $8 \\times 8$ grid of cells with one number in each cell. Let $a_{i,j}$ be the number placed in the cell in row $i$... |
aime_26 | mathematics | {
"ground_truth": "161",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Triangle $\\triangle ABC$ lies in plane $\\mathcal P$ with $AB = 6, AC = 4,$ and $\\angle BAC = 90^\\circ.$ Let $D$ be the reflection across $\\overline{BC}$ of the centroid of $... |
aime_26 | mathematics | {
"ground_truth": "39",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each positive integer $r$ less than $502,$ define\n\\[\nS_r=\\sum_{m\\ge 0}\\dbinom{10000}{502n+r},\n\\]\nwhere $\\binom{10000}{n}$ is defined to be $0$ when $n>10000.$ That ... |
aime_26 | mathematics | {
"ground_truth": "681",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In an equiangular pentagon, the sum of the squares of the side lengths equals $308,$ and the sum of the squares of the diagonal lengths equals $800.$ The square of the perimeter ... |
aime_26 | mathematics | {
"ground_truth": "83",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a, b,$ and $n$ be positive integers with both $a$ and $b$ greater than or equal to $2$ and less than or equal to $2n$. Define an $a \\times b$ cell loop in a $2n \\times 2n$... |
aime_26 | mathematics | {
"ground_truth": "178",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of the $10$th terms of all arithmetic sequences of integers that have first term equal to $4$ and include both $24$ and $34$ as terms.",
"role": "user"
}
] |
aime_26 | mathematics | {
"ground_truth": "243",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The figure below shows a grid of $10$ squares in a row. Each square has a diagonal connecting its lower left vertex to its upper right vertex. A bug moves along the line segments... |
aime_26 | mathematics | {
"ground_truth": "503",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABCDE$ be a nonconvex pentagon with internal angles $\\angle A = \\angle E = 90^\\circ$ and $\\angle B = \\angle D = 45^\\circ.$ Suppose that $DE < AB, AE = 20, BC = 14\\sqr... |
aime_26 | mathematics | {
"ground_truth": "279",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each positive integer $n$ let $f(n)$ be the value of the base-ten numeral $n$ viewed in base $b$, where $b$ is the least integer greater than the greatest digit in $n$. For e... |
aime_26 | mathematics | {
"ground_truth": "190",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An urn contains $n$ marbles. Each marble is either red or blue, and there are at least $7$ marbles of each color. When $7$ marbles are drawn randomly from the urn without replace... |
aime_26 | mathematics | {
"ground_truth": "50",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of all real numbers $r$ such that there is at least one point where the circle with radius $r$ centered at $(4, 39)$ is tangent to the parabola with equation $2y = x... |
aime_26 | mathematics | {
"ground_truth": "754",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A standard fair six-sided die is rolled repeatedly. Each time the die reads 1 or 2, Alice gets a coin; each time it reads 3 or 4, Bob gets a coin; and each time it reads 5 or 6, ... |
aime_26 | mathematics | {
"ground_truth": "245",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Isosceles triangle $\\triangle ABC$ has $AB = BC.$ Let $I$ be the incenter of $\\triangle ABC.$ The perimeters of $\\triangle ABC$ and $\\triangle AIC$ are in the ratio $125:6,$ ... |
aime_26 | mathematics | {
"ground_truth": "669",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ denote the value of the infinite sum\n\\[\n\\frac{1}{9} + \\frac{1}{99} + \\frac{1}{999} + \\frac{1}{9999} + \\cdots\n\\]\nFind the remainder when the greatest integer l... |
aime_26 | mathematics | {
"ground_truth": "850",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ be a triangle with $D$ on $\\overline{BC}$ such that $\\overline{AD}$ bisects $\\angle BAC.$ Let $\\omega$ be the circle that passes through $A$ and is tange... |
aime_26 | mathematics | {
"ground_truth": "132",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the greatest integer $n$ such that the cubic polynomial\n\\[\nx^{3} - \\displaystyle\\frac{n}{6}x^{2} + (n - 11)x - 400\n\\]\nhas roots $\\alpha^{2}$, $\\beta^{2}$, and $\\g... |
aime_26 | mathematics | {
"ground_truth": "223",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a tetrahedron with two isosceles triangle faces with side lengths $5\\sqrt{10}, 5\\sqrt{10},$ and $10$ and two isosceles triangle faces with side lengths $5\\sqrt{10}, 5... |
aime_26 | mathematics | {
"ground_truth": "107",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Call finite sets of integers $S$ and $T$ cousins if\n- $S$ and $T$ have the same number of elements,\n- $S$ and $T$ are disjoint, and\n- the elements of $S$ can be paired with th... |
aime_26 | mathematics | {
"ground_truth": "157",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For integers $a$ and $b,$ let $a \\circ b = a - b$ if $a$ is odd and $b$ is even, and $a+b$ otherwise. Find the number of sequences $a_1, a_2, a_3, \\ldots, a_n$ of positive inte... |
aime_26 | mathematics | {
"ground_truth": "393",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of ordered 7-tuples $(a_1, a_2, a_3, \\ldots, a_7)$ having the following properties:\n- $a_k \\in \\{1,2,3\\}$ for all $k.$\n- $a_1+a_2+a_3+a_4+a_5+a_6+a_7$ is a ... |
aime_26 | mathematics | {
"ground_truth": "277",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Patrick started walking at a constant rate along a straight road from school to the park. One hour after Patrick left, Tanya started running along the same road from school to th... |
aime_26 | mathematics | {
"ground_truth": "62",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of positive integer palindromes written in base $10$ with no zero digits, and whose digits add up to $13$. For example, $42124$ has these properties. Recall that ... |
aime_26 | mathematics | {
"ground_truth": "79",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A hemisphere with radius $200$ sits on top of a horizontal circular disk with radius $200,$ and the hemisphere and disk have the same center. Let $\\mathcal T$ be the region of p... |
aime_26 | mathematics | {
"ground_truth": "70",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of integers less than or equal to 100 that are equal to $a+b+ab$ for some choice of distinct positive integers a and b.",
"role": "user"
}
] |
aime_26 | mathematics | {
"ground_truth": "65",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\\theta$ around point $B$ to point $A^\\prime$. Then... |
aime_26 | mathematics | {
"ground_truth": "441",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A real number $x$ satisfies $\\sqrt[20]{x^{\\log_{2026}x}}=26x$. What is the number of positive divisors of the product of all possible positive values of $x$?",
"role": "use... |
aime_26 | mathematics | {
"ground_truth": "396",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "\nFind the number of functions $\\pi$ mapping the set $A =\\{1,2,3,4,5,6\\}$ onto $A$ such that for every $a \\in A,$\n\\[\n\\pi(\\pi(\\pi(\\pi(\\pi(\\pi(a)))))) = a.\n\\]",
... |
aime_26 | mathematics | {
"ground_truth": "244",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $N$ be the number of positive integer divisors of $17017^{17}$ that leave a remainder of $5$ when divided by $12$. Find the remainder when $N$ is divided by $1000$.",
"ro... |
aime_26 | mathematics | {
"ground_truth": "29",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Joanne has a blank fair six-sided die and six stickers each displaying a different integer from 1 to 6. Joanne rolls the die and then places the sticker labeled 1 on the top face... |
aime_26 | mathematics | {
"ground_truth": "156",
"style": "rule"
} | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ have side lengths $AB = 13, BC = 14,$ and $CA = 15.$ Triangle $\\triangle A'B'C'$ is obtained by rotating $\\triangle ABC$ about its circumcenter so that ${}... |
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