prompt listlengths 1 1 | reward_model dict | data_source stringclasses 7
values | ability stringclasses 1
value | extra_info dict |
|---|---|---|---|---|
[
{
"content": "Find the number of triples of nonnegative integers \\((a,b,c)\\) satisfying \\(a + b + c = 300\\) and\n\\begin{equation*}\na^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.\n\\end{equation*}",
"role": "user"
}
] | {
"ground_truth": "601",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 17,
"problem": "Find the number of triples of nonnegative integers \\((a,b,c)\\) satisfying \\(a + b + c = 300\\) and\n\\begin{equation*}\na^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.\n\\end{equation*}"
} |
[
{
"content": "Let \\(O=(0,0)\\), \\(A=\\left(\\tfrac{1}{2},0\\right)\\), and \\(B=\\left(0,\\tfrac{\\sqrt{3}}{2}\\right)\\) be points in the coordinate plane. Let \\(\\mathcal{F}\\) be the family of segments \\(\\overline{PQ}\\) of unit length lying in the first quadrant with \\(P\\) on the \\(x\\)-axis and \\(... | {
"ground_truth": "023",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 18,
"problem": "Let \\(O=(0,0)\\), \\(A=\\left(\\tfrac{1}{2},0\\right)\\), and \\(B=\\left(0,\\tfrac{\\sqrt{3}}{2}\\right)\\) be points in the coordinate plane. Let \\(\\mathcal{F}\\) be the family of segments \\(\\overline{PQ}\\) of unit length lying in the first quadrant with \\(P\\) on the \\(x\\)-axi... |
[
{
"content": "Let $\\omega\\neq 1$ be a 13th root of unity. Find the remainder when\n\\[\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})\\]\nis divided by 1000.",
"role": "user"
}
] | {
"ground_truth": "321",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 19,
"problem": "Let $\\omega\\neq 1$ be a 13th root of unity. Find the remainder when\n\\[\\prod_{k=0}^{12}(2-2\\omega^k+\\omega^{2k})\\]\nis divided by 1000."
} |
[
{
"content": "Let \\(b\\ge 2\\) be an integer. Call a positive integer \\(n\\) \\(b\\text-\\textit{eautiful}\\) if it has exactly two digits when expressed in base \\(b\\) and these two digits sum to \\(\\sqrt n\\). For example, \\(81\\) is \\(13\\text-\\textit{eautiful}\\) because \\(81 = \\underline{6} \\ \... | {
"ground_truth": "211",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 20,
"problem": "Let \\(b\\ge 2\\) be an integer. Call a positive integer \\(n\\) \\(b\\text-\\textit{eautiful}\\) if it has exactly two digits when expressed in base \\(b\\) and these two digits sum to \\(\\sqrt n\\). For example, \\(81\\) is \\(13\\text-\\textit{eautiful}\\) because \\(81 = \\underlin... |
[
{
"content": "Find the number of rectangles that can be formed inside a fixed regular dodecagon ($12$-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles.\n[asy] unitsize(0.6 inch); for(int i=0; i<360; i+=30) { dot(dir(i),... | {
"ground_truth": "315",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 21,
"problem": "Find the number of rectangles that can be formed inside a fixed regular dodecagon ($12$-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles.\n[asy] unitsize(0.6 inch); for(int i=0; i<360; i+=30) { do... |
[
{
"content": "A list of positive integers has the following properties:\n$\\bullet$ The sum of the items in the list is $30$.\n$\\bullet$ The unique mode of the list is $9$.\n$\\bullet$ The median of the list is a positive integer that does not appear in the list itself.\nFind the sum of the squares of all the ... | {
"ground_truth": "236",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 22,
"problem": "A list of positive integers has the following properties:\n$\\bullet$ The sum of the items in the list is $30$.\n$\\bullet$ The unique mode of the list is $9$.\n$\\bullet$ The median of the list is a positive integer that does not appear in the list itself.\nFind the sum of the squares of... |
[
{
"content": "Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is $999$, and the sum of the three numbers formed by reading top to bottom is $99$. The grid below is an example of such an arrangement because $8+991=999$ and $9+... | {
"ground_truth": "045",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 23,
"problem": "Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is $999$, and the sum of the three numbers formed by reading top to bottom is $99$. The grid below is an example of such an arrangement because $8+991=999... |
[
{
"content": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({z \\over xy}\\right) = {1 \\over 4}\\]\nThen the value of $\\left|\\log_2(x^... | {
"ground_truth": "033",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 24,
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({z \\over xy}\\right) = {1 \\over 4}\\]\nThen the value of $\\left|\... |
[
{
"content": "Let ABCDEF be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments AB, CD, and EF has side lengths 200, 240, and 300. Find the side length of the hexagon.",
"role": "user"
}
] | {
"ground_truth": "080",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 25,
"problem": "Let ABCDEF be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments AB, CD, and EF has side lengths 200, 240, and 300. Find the side length of the hexagon."
} |
[
{
"content": "Alice chooses a set $A$ of positive integers. Then Bob lists all finite nonempty sets $B$ of positive integers with the property that the maximum element of $B$ belongs to $A$. Bob's list has 2024 sets. Find the sum of the elements of A.",
"role": "user"
}
] | {
"ground_truth": "055",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 26,
"problem": "Alice chooses a set $A$ of positive integers. Then Bob lists all finite nonempty sets $B$ of positive integers with the property that the maximum element of $B$ belongs to $A$. Bob's list has 2024 sets. Find the sum of the elements of A."
} |
[
{
"content": "Let $N$ be the greatest four-digit positive integer with the property that whenever one of its digits is changed to $1$, the resulting number is divisible by $7$. Let $Q$ and $R$ be the quotient and remainder, respectively, when $N$ is divided by $1000$. Find $Q+R$.",
"role": "user"
}
] | {
"ground_truth": "699",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 27,
"problem": "Let $N$ be the greatest four-digit positive integer with the property that whenever one of its digits is changed to $1$, the resulting number is divisible by $7$. Let $Q$ and $R$ be the quotient and remainder, respectively, when $N$ is divided by $1000$. Find $Q+R$."
} |
[
{
"content": "Torus $T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane of the circle that is a distance $6$ from the center of the circle (so like a donut). Let $S$ be a sphere with a radius $11$. When $T$ rests on the outside of $S$, it is externally tangent to $S$ al... | {
"ground_truth": "127",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 28,
"problem": "Torus $T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane of the circle that is a distance $6$ from the center of the circle (so like a donut). Let $S$ be a sphere with a radius $11$. When $T$ rests on the outside of $S$, it is externally tangent ... |
[
{
"content": "There is a collection of $25$ indistinguishable white chips and $25$ indistinguishable black chips. Find the number of ways to place some of these chips in the $25$ unit cells of a $5\\times5$ grid such that: \n\neach cell contains at most one chip\nall chips in the same row and all chips in the s... | {
"ground_truth": "902",
"style": "rule-lighteval/MATH_v2"
} | aime24 | MATH | {
"index": 29,
"problem": "There is a collection of $25$ indistinguishable white chips and $25$ indistinguishable black chips. Find the number of ways to place some of these chips in the $25$ unit cells of a $5\\times5$ grid such that: \n\neach cell contains at most one chip\nall chips in the same row and all chips... |
[
{
"content": "Find the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$.",
"role": "user"
}
] | {
"ground_truth": "70",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 0,
"problem": "Find the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$."
} |
[
{
"content": "On $\\triangle ABC$ points $A,D,E$, and $B$ lie that order on side $\\overline{AB}$ with $AD=4, DE=16$, and $EB=8$. Points $A,F,G$, and $C$ lie in that order on side $\\overline{AC}$ with $AF=13, FG=52$, and $GC=26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G... | {
"ground_truth": "588",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 1,
"problem": "On $\\triangle ABC$ points $A,D,E$, and $B$ lie that order on side $\\overline{AB}$ with $AD=4, DE=16$, and $EB=8$. Points $A,F,G$, and $C$ lie in that order on side $\\overline{AC}$ with $AF=13, FG=52$, and $GC=26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflecti... |
[
{
"content": "The 9 members of a baseball team went to an ice cream parlor after their game. Each player had a singlescoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose va... | {
"ground_truth": "16",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 2,
"problem": "The 9 members of a baseball team went to an ice cream parlor after their game. Each player had a singlescoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who ... |
[
{
"content": "Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$, inclusive, such that $12x^{2}-xy-6y^{2}=0$.",
"role": "user"
}
] | {
"ground_truth": "117",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 3,
"problem": "Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$, inclusive, such that $12x^{2}-xy-6y^{2}=0$."
} |
[
{
"content": "There are $8!=40320$ eight-digit positive integers that use each of the digits $1,2,3,4,5,6,7,8$ exactly once. Let $N$ be the number of these integers that are divisible by 22. Find the difference between $N$ and 2025.",
"role": "user"
}
] | {
"ground_truth": "279",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 4,
"problem": "There are $8!=40320$ eight-digit positive integers that use each of the digits $1,2,3,4,5,6,7,8$ exactly once. Let $N$ be the number of these integers that are divisible by 22. Find the difference between $N$ and 2025."
} |
[
{
"content": "An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is 3, and the area of the trapezoid is 72. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \\neq s$. Find $r^{2}+s^{2}$.",
"role": "user"
}
] | {
"ground_truth": "504",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 5,
"problem": "An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is 3, and the area of the trapezoid is 72. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \\neq s$. Find $r^{2}+s^{2}$."
} |
[
{
"content": "The 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 those six words are listed alphabetically. For example, a possible result is $AB,CJ... | {
"ground_truth": "821",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 6,
"problem": "The 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 those six words are listed alphabetically. For example, a possible result i... |
[
{
"content": "Let $k$ be real numbers such that the system $|25+20i-z|=5$ and $|z-4-k|=|z-3i-k|$ has exactly one complex solution $z$. The sum of all possible values of $k$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Here $i=\\sqrt{-1}$.",
"role": ... | {
"ground_truth": "77",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 7,
"problem": "Let $k$ be real numbers such that the system $|25+20i-z|=5$ and $|z-4-k|=|z-3i-k|$ has exactly one complex solution $z$. The sum of all possible values of $k$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Here $i=\\sqrt{-1}$."
} |
[
{
"content": "The parabola with equation $y=x^{2}-4$ is rotated $60^{\\circ}$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\\frac{a-\\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c... | {
"ground_truth": "62",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 8,
"problem": "The parabola with equation $y=x^{2}-4$ is rotated $60^{\\circ}$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\\frac{a-\\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a... |
[
{
"content": "The 27 cells of a $3\\times9$ grid are filled in using the numbers 1 through 9 so that each row contains 9 different numbers, and each of the three $3\\times3$ blocks heavily outlined in the example below contains 9 different numbers, as in the first three rows of a Sudoku puzzle. \n | 4 | 2 | 8 |... | {
"ground_truth": "81",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 9,
"problem": "The 27 cells of a $3\\times9$ grid are filled in using the numbers 1 through 9 so that each row contains 9 different numbers, and each of the three $3\\times3$ blocks heavily outlined in the example below contains 9 different numbers, as in the first three rows of a Sudoku puzzle. \n | 4 |... |
[
{
"content": "A piecewise linear periodic function is defined by $f(x)=\\begin{cases}x&\\text{if }x\\in[-1,1)\\\\2-x&\\text{if }x\\in[1,3)\\end{cases}$ and $f(x+4)=f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern. The parabola $x=34y^2$ intersects the graph of $f(x)$ at finitely many... | {
"ground_truth": "259",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 10,
"problem": "A piecewise linear periodic function is defined by $f(x)=\\begin{cases}x&\\text{if }x\\in[-1,1)\\\\2-x&\\text{if }x\\in[1,3)\\end{cases}$ and $f(x+4)=f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern. The parabola $x=34y^2$ intersects the graph of $f(x)$ at fini... |
[
{
"content": "The set of points in 3-dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities $x-yz<y-zx<z-xy$ forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\\sqrt{b... | {
"ground_truth": "510",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 11,
"problem": "The set of points in 3-dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities $x-yz<y-zx<z-xy$ forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $... |
[
{
"content": "Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws 25 more line segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting those two poin... | {
"ground_truth": "204",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 12,
"problem": "Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws 25 more line segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting those... |
[
{
"content": "Let $ABCDE$ be a convex pentagon with $AB=14, BC=7, CD=24, DE=13, EA=26,$ and $\\angle B=\\angle E=60^\\circ$. For each point $X$ in the plane, define $f(X)=AX+BX+CX+DX+EX$. The least possible value of $f(X)$ can be expressed as $m+n\\sqrt{p}$, where $m$ and $n$ are positive integers and $p$ is no... | {
"ground_truth": "60",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 13,
"problem": "Let $ABCDE$ be a convex pentagon with $AB=14, BC=7, CD=24, DE=13, EA=26,$ and $\\angle B=\\angle E=60^\\circ$. For each point $X$ in the plane, define $f(X)=AX+BX+CX+DX+EX$. The least possible value of $f(X)$ can be expressed as $m+n\\sqrt{p}$, where $m$ and $n$ are positive integers and ... |
[
{
"content": "Let $N$ denote the number of ordered triples of positive integers $(a,b,c)$ such that $a,b,c\\leq3^6$ and $a^3+b^3+c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.",
"role": "user"
}
] | {
"ground_truth": "735",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 14,
"problem": "Let $N$ denote the number of ordered triples of positive integers $(a,b,c)$ such that $a,b,c\\leq3^6$ and $a^3+b^3+c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$."
} |
[
{
"content": "Six points $ A, B, C, D, E, $ and $ F $ lie in a straight line in that order. Suppose that $ G $ is a point not on the line and that $ AC = 26 $, $ BD = 22 $, $ CE = 31 $, $ DF = 33 $, $ AF = 73 $, $ CG = 40 $, and $ DG = 30 $. Find the area of $ \\triangle BGE $.",
"role": "user"
}
] | {
"ground_truth": "468",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 15,
"problem": "Six points $ A, B, C, D, E, $ and $ F $ lie in a straight line in that order. Suppose that $ G $ is a point not on the line and that $ AC = 26 $, $ BD = 22 $, $ CE = 31 $, $ DF = 33 $, $ AF = 73 $, $ CG = 40 $, and $ DG = 30 $. Find the area of $ \\triangle BGE $."
} |
[
{
"content": "Find the sum of all positive integers $ n $ such that $ n + 2 $ divides the product $ 3(n + 3)(n^2 + 9) $.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 16,
"problem": "Find the sum of all positive integers $ n $ such that $ n + 2 $ divides the product $ 3(n + 3)(n^2 + 9) $."
} |
[
{
"content": "Four unit squares form a $2 \\times 2$ grid. Each of the 12 unit line segments forming the sides of the squares is colored either red or blue in such a way that each unit square has 2 red sides and 2 blue sides. Find the number of such colorings.",
"role": "user"
}
] | {
"ground_truth": "82",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 17,
"problem": "Four unit squares form a $2 \\times 2$ grid. Each of the 12 unit line segments forming the sides of the squares is colored either red or blue in such a way that each unit square has 2 red sides and 2 blue sides. Find the number of such colorings."
} |
[
{
"content": "The product $ \\prod_{k=4}^{63} \\frac{\\log_k(5^{k^2-1})}{\\log_{k+1}(5^{k^2-4})} = \\frac{\\log_4(5^{15})}{\\log_5(5^{12})} \\cdot \\frac{\\log_5(5^{24})}{\\log_6(5^{21})} \\cdot \\frac{\\log_6(5^{35})}{\\log_7(5^{32})} \\cdots \\frac{\\log_{63}(5^{3968})}{\\log_{64}(5^{3965})} $ is equal to $ \... | {
"ground_truth": "106",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 18,
"problem": "The product $ \\prod_{k=4}^{63} \\frac{\\log_k(5^{k^2-1})}{\\log_{k+1}(5^{k^2-4})} = \\frac{\\log_4(5^{15})}{\\log_5(5^{12})} \\cdot \\frac{\\log_5(5^{24})}{\\log_6(5^{21})} \\cdot \\frac{\\log_6(5^{35})}{\\log_7(5^{32})} \\cdots \\frac{\\log_{63}(5^{3968})}{\\log_{64}(5^{3965})} $ is equ... |
[
{
"content": "Suppose $ \\triangle ABC $ has angles $ \\angle BAC = 84^\\circ $, $ \\angle ABC = 60^\\circ $, and $ \\angle ACB = 36^\\circ $. Let $ D, E, $ and $ F $ be the midpoints of sides $ \\overline{BC} $, $ \\overline{AC} $, and $ \\overline{AB} $, respectively. The circumcircle of $ \\triangle DEF $ in... | {
"ground_truth": "336^\\circ",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 19,
"problem": "Suppose $ \\triangle ABC $ has angles $ \\angle BAC = 84^\\circ $, $ \\angle ABC = 60^\\circ $, and $ \\angle ACB = 36^\\circ $. Let $ D, E, $ and $ F $ be the midpoints of sides $ \\overline{BC} $, $ \\overline{AC} $, and $ \\overline{AB} $, respectively. The circumcircle of $ \\triangle... |
[
{
"content": "Circle $\\omega_1$ with radius 6 centered at point $A$ is internally tangent at point $B$ to circle $\\omega_2$ with radius 15. Points $C$ and $D$ lie on $\\omega_2$ such that $\\overline{BC}$ is a diameter of $\\omega_2$ and $\\overline{BC} \\perp \\overline{AD}$. The rectangle $EFGH$ is inscribe... | {
"ground_truth": "293",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 20,
"problem": "Circle $\\omega_1$ with radius 6 centered at point $A$ is internally tangent at point $B$ to circle $\\omega_2$ with radius 15. Points $C$ and $D$ lie on $\\omega_2$ such that $\\overline{BC}$ is a diameter of $\\omega_2$ and $\\overline{BC} \\perp \\overline{AD}$. The rectangle $EFGH$ is... |
[
{
"content": "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 positiv... | {
"ground_truth": "237",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 21,
"problem": "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 prim... |
[
{
"content": "From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $ N $ cents, where $ N $ is a positive integer. He uses the so-called **greedy algorithm**, successively choosing the coin of greatest value that does not... | {
"ground_truth": "610",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 22,
"problem": "From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $ N $ cents, where $ N $ is a positive integer. He uses the so-called **greedy algorithm**, successively choosing the coin of greatest value that... |
[
{
"content": "There are $ n $ values of $ x $ in the interval $ 0 < x < 2\\pi $ where $ f(x) = \\sin(7\\pi \\cdot \\sin(5x)) = 0 $. For $ t $ of these $ n $ values of $ x $, the graph of $ y = f(x) $ is tangent to the $ x $-axis. Find $ n + t $.",
"role": "user"
}
] | {
"ground_truth": "149",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 23,
"problem": "There are $ n $ values of $ x $ in the interval $ 0 < x < 2\\pi $ where $ f(x) = \\sin(7\\pi \\cdot \\sin(5x)) = 0 $. For $ t $ of these $ n $ values of $ x $, the graph of $ y = f(x) $ is tangent to the $ x $-axis. Find $ n + t $."
} |
[
{
"content": "Sixteen 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 16 chairs that could be selected. Find the remainder when $ N $ is divided by 1000.",
"role": "user"
}
] | {
"ground_truth": "907",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 24,
"problem": "Sixteen 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 16 chairs that could be selected. Find the remainder when $ N $ is divided by 1000."
} |
[
{
"content": "Let $ S $ be the set of vertices of a regular 24-gon. Find the number of ways to draw 12 segments of equal lengths so that each vertex in $ S $ is an endpoint of exactly one of the 12 segments.",
"role": "user"
}
] | {
"ground_truth": "113",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 25,
"problem": "Let $ S $ be the set of vertices of a regular 24-gon. Find the number of ways to draw 12 segments of equal lengths so that each vertex in $ S $ is an endpoint of exactly one of the 12 segments."
} |
[
{
"content": "Let $ A_1A_2 \\ldots A_{11} $ be an 11-sided non-convex simple polygon with the following properties:\n* The area of $ A_iA_1A_{i+1} $ is 1 for each $ 2 \\leq i \\leq 10 $,\n* $ \\cos(\\angle A_iA_1A_{i+1}) = \\frac{12}{13} $ for each $ 2 \\leq i \\leq 10 $,\n* The perimeter of $ A_1A_2 \\ldots A_... | {
"ground_truth": "19",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 26,
"problem": "Let $ A_1A_2 \\ldots A_{11} $ be an 11-sided non-convex simple polygon with the following properties:\n* The area of $ A_iA_1A_{i+1} $ is 1 for each $ 2 \\leq i \\leq 10 $,\n* $ \\cos(\\angle A_iA_1A_{i+1}) = \\frac{12}{13} $ for each $ 2 \\leq i \\leq 10 $,\n* The perimeter of $ A_1A_2 \... |
[
{
"content": "Let the sequence of rationals $ x_1, x_2, \\ldots $ be defined such that $ x_1 = \\frac{25}{11} $ and\n$ x_{k+1} = \\frac{1}{3} \\left( x_k + \\frac{1}{x_k} - 1 \\right). $\n$ x_{2025} $ can be expressed as $ \\frac{m}{n} $ for relatively prime positive integers $ m $ and $ n $. Find the remainder... | {
"ground_truth": "248",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 27,
"problem": "Let the sequence of rationals $ x_1, x_2, \\ldots $ be defined such that $ x_1 = \\frac{25}{11} $ and\n$ x_{k+1} = \\frac{1}{3} \\left( x_k + \\frac{1}{x_k} - 1 \\right). $\n$ x_{2025} $ can be expressed as $ \\frac{m}{n} $ for relatively prime positive integers $ m $ and $ n $. Find the ... |
[
{
"content": "Let $ \\triangle ABC $ be a right triangle with $ \\angle A = 90^\\circ $ and $ BC = 38 $. There exist points $ K $ and $ L $ inside the triangle such that $ AK = AL = BK = CL = KL = 14. $ The area of the quadrilateral $ BKLC $ can be expressed as $ n \\sqrt{3} $ for some positive integer $ n $. F... | {
"ground_truth": "104",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 28,
"problem": "Let $ \\triangle ABC $ be a right triangle with $ \\angle A = 90^\\circ $ and $ BC = 38 $. There exist points $ K $ and $ L $ inside the triangle such that $ AK = AL = BK = CL = KL = 14. $ The area of the quadrilateral $ BKLC $ can be expressed as $ n \\sqrt{3} $ for some positive integer... |
[
{
"content": "There are exactly three positive real numbers $ k $ such that the function\n$ f(x) = \\frac{(x - 18)(x - 72)(x - 98)(x - k)}{x} $\ndefined over the positive real numbers achieves its minimum value at exactly two positive real numbers $ x $. Find the sum of these three values of $ k $.",
"role"... | {
"ground_truth": "240",
"style": "rule-lighteval/MATH_v2"
} | aime25 | MATH | {
"index": 29,
"problem": "There are exactly three positive real numbers $ k $ such that the function\n$ f(x) = \\frac{(x - 18)(x - 72)(x - 98)(x - k)}{x} $\ndefined over the positive real numbers achieves its minimum value at exactly two positive real numbers $ x $. Find the sum of these three values of $ k $."
} |
[
{
"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 the park. One hour after Tanya left, Jose started bicycling along the same road from school to the park. Tanya ran at a co... | {
"ground_truth": "277",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 0,
"problem": "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 the park. One hour after Tanya left, Jose started bicycling along the same road from school to the park. Tanya ran... |
[
{
"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 a palindrome is a number whose representation reads the same from left to right as from right to left.",
"role": "us... | {
"ground_truth": "62",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 1,
"problem": "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 a palindrome is a number whose representation reads the same from left to right as from right to left."
} |
[
{
"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 points P in the disk such that a sphere of radius $42$ can be placed on top of the disk at $P$ and lie completely inside ... | {
"ground_truth": "79",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 2,
"problem": "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 points P in the disk such that a sphere of radius $42$ can be placed on top of the disk at $P$ and lie completely... |
[
{
"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"
}
] | {
"ground_truth": "70",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 3,
"problem": "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."
} |
[
{
"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 $B$ is rotated in the plane clockwise through angle $\\theta$ around point $A^\\prime$ to point $B^\\prime$. Suppose th... | {
"ground_truth": "65",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 4,
"problem": "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 $B$ is rotated in the plane clockwise through angle $\\theta$ around point $A^\\prime$ to point $B^\\prime$. Su... |
[
{
"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": "user"
}
] | {
"ground_truth": "441",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 5,
"problem": "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 $x$?"
} |
[
{
"content": "Find 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\\]",
"role": "user"
}
] | {
"ground_truth": "396",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 6,
"problem": "Find 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\\]"
} |
[
{
"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 $1000$.",
"role": "user"
}
] | {
"ground_truth": "244",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 7,
"problem": "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 $1000$."
} |
[
{
"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 of the die. She then rolls the die again, places the sticker labeled 2 on the top face, and continues this process to p... | {
"ground_truth": "29",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 8,
"problem": "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 of the die. She then rolls the die again, places the sticker labeled 2 on the top face, and continues this proc... |
[
{
"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 ${}\\overline{AC}$ is perpendicular $\\overline{BC},$ with $A'$ and $B$ not on the same side of line $B'C'.$ Find the integ... | {
"ground_truth": "156",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 9,
"problem": "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 ${}\\overline{AC}$ is perpendicular $\\overline{BC},$ with $A'$ and $B$ not on the same side of line $B'C'.$ Find t... |
[
{
"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 $i$ and column $j,$ and let $M$ be the sum of the absolute differences between adjacent cells. That is,\n\\[\nM = \\sum^8_{... | {
"ground_truth": "896",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 10,
"problem": "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 $i$ and column $j,$ and let $M$ be the sum of the absolute differences between adjacent cells. That is,\n\\[\nM = ... |
[
{
"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 $\\triangle ABC. {}$ Four spheres, all on the same side of $\\mathcal P,$ have radii $1, 2, 3,$ and $r$ and are tangent t... | {
"ground_truth": "161",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 11,
"problem": "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 $\\triangle ABC. {}$ Four spheres, all on the same side of $\\mathcal P,$ have radii $1, 2, 3,$ and $r$ and are ... |
[
{
"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 is, $S_r$ is the sum of all binomial coefficients of the form $\\binom{10000}{k}$ for which $0\\le k\\le 10000$ and $k-r... | {
"ground_truth": "39",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 12,
"problem": "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 is, $S_r$ is the sum of all binomial coefficients of the form $\\binom{10000}{k}$ for which $0\\le k\\le 10000$... |
[
{
"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 of the pentagon can be expressed as $m \\sqrt n,$ where $m$ and $n$ are positive integers and $n$ is not divisible by th... | {
"ground_truth": "681",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 13,
"problem": "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 of the pentagon can be expressed as $m \\sqrt n,$ where $m$ and $n$ are positive integers and $n$ is not divisi... |
[
{
"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 $2n \\times 2n$ grid of cells to be the $2a + 2b - 4$ cells that surround an $(a - 2) \\times (b - 2)$ (possibly empty) rectangle of ce... | {
"ground_truth": "83",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 14,
"problem": "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 $2n \\times 2n$ grid of cells to be the $2a + 2b - 4$ cells that surround an $(a - 2) \\times (b - 2)$ (possibly empty) rectan... |
[
{
"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"
}
] | {
"ground_truth": "178",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 15,
"problem": "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."
} |
[
{
"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 from vertex to vertex, never traversing the same segment twice and never moving from right to left along a horizontal o... | {
"ground_truth": "243",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 16,
"problem": "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 from vertex to vertex, never traversing the same segment twice and never moving from right to left along a hor... |
[
{
"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 $DE < AB, AE = 20, BC = 14\\sqrt2,$ and points $B,C,$ and $D$ lie on the same side of line $AE.$ Suppose further that $AB$ is an integer with $AB < 202... | {
"ground_truth": "503",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 17,
"problem": "Let $ABCDE$ be a nonconvex pentagon with internal angles $\\angle A = \\angle E = 90^\\circ$ and $\\angle B = \\angle D = 45^\\circ.$ Suppose that $DE < AB, AE = 20, BC = 14\\sqrt2,$ and points $B,C,$ and $D$ lie on the same side of line $AE.$ Suppose further that $AB$ is an integer with ... |
[
{
"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 example, if $n=72$, then $b=8$, and $72$ as a numeral in base $8$ equals $7\\cdot 8+2=58$; therefore $f(72)=58$. Find the... | {
"ground_truth": "279",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 18,
"problem": "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 example, if $n=72$, then $b=8$, and $72$ as a numeral in base $8$ equals $7\\cdot 8+2=58$; therefore $f(72)=58$.... |
[
{
"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 replacement, the probability that exactly $4$ of them are red equals the probability that exactly $5$ of them are red. Find the... | {
"ground_truth": "190",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 19,
"problem": "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 replacement, the probability that exactly $4$ of them are red equals the probability that exactly $5$ of them are red.... |
[
{
"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 $2y = x^2 - 8x + 12.$",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 20,
"problem": "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 $2y = x^2 - 8x + 12.$"
} |
[
{
"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, Carol gets a coin. The probability that Alice and Bob each receive at least two coins before Carol receives any coins ca... | {
"ground_truth": "754",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 21,
"problem": "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, Carol gets a coin. The probability that Alice and Bob each receive at least two coins before Carol receives any... |
[
{
"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 $125:6,$ and all the sides of both triangles have integer lengths. Find the minimum possible value of $AB.$",
"role": "user"
... | {
"ground_truth": "245",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 22,
"problem": "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 $125:6,$ and all the sides of both triangles have integer lengths. Find the minimum possible value of $AB.$"
} |
[
{
"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 less than or equal to $10^{100} S$ is divided by $1000.$",
"role": "user"
}
] | {
"ground_truth": "669",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 23,
"problem": "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 less than or equal to $10^{100} S$ is divided by $1000.$"
} |
[
{
"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 tangent to segment $\\overline{BC}$ at $D.$ Let $E \\neq A$ and $F \\neq A$ be the intersections of $\\omega$ with segments $... | {
"ground_truth": "850",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 24,
"problem": "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 tangent to segment $\\overline{BC}$ at $D.$ Let $E \\neq A$ and $F \\neq A$ be the intersections of $\\omega$ with s... |
[
{
"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 $\\gamma^{2}$, where $\\alpha$, $\\beta$, and $\\gamma$ are complex numbers, and there are exactly seven different possible ... | {
"ground_truth": "132",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 25,
"problem": "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 $\\gamma^{2}$, where $\\alpha$, $\\beta$, and $\\gamma$ are complex numbers, and there are exactly seven different ... |
[
{
"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 $5\\sqrt{10}, 5\\sqrt{10},$ and $18.$ The four vertices of the tetrahedron lie on a sphere with center $S,$ and the four faces of the t... | {
"ground_truth": "223",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 26,
"problem": "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 $5\\sqrt{10}, 5\\sqrt{10},$ and $18.$ The four vertices of the tetrahedron lie on a sphere with center $S,$ and the four faces... |
[
{
"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 the elements of $T$ so that the elements in each pair differ by exactly $1$.\nFor example, $\\{1,2,5\\}$ and $\\{0,3,4\\}$... | {
"ground_truth": "107",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 27,
"problem": "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 the elements of $T$ so that the elements in each pair differ by exactly $1$.\nFor example, $\\{1,2,5\\}$ and $\\{... |
[
{
"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 integers such that\n\\[\na_1 + a_2 + a_3 + \\cdots + a_n = 12 \\quad \\text{and} \\quad a_1 \\circ a_2 \\circ a_3 \\circ \\c... | {
"ground_truth": "157",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 28,
"problem": "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 integers such that\n\\[\na_1 + a_2 + a_3 + \\cdots + a_n = 12 \\quad \\text{and} \\quad a_1 \\circ a_2 \\circ a_3 \... |
[
{
"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 multiple of $3.$\n- $a_1a_2 a_4 + a_2a_3a_5 + a_3a_4 a_6 + a_4 a_5 a_7 + a_5 a_6 a_1 + a_6 a_7 a_2 + a_7 a_1 a_3$ is a m... | {
"ground_truth": "393",
"style": "rule-lighteval/MATH_v2"
} | aime26 | MATH | {
"index": 29,
"problem": "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 multiple of $3.$\n- $a_1a_2 a_4 + a_2a_3a_5 + a_3a_4 a_6 + a_4 a_5 a_7 + a_5 a_6 a_1 + a_6 a_7 a_2 + a_7 a_1 a_... |
[
{
"content": "One hundred concentric circles are labelled $C_{1}, C_{2}, C_{3}, \\ldots, C_{100}$. Each circle $C_{n}$ is inscribed within an equilateral triangle whose vertices are points on $C_{n+1}$. Given $C_{1}$ has a radius of $1$, what is the radius of $C_{100}$ ?",
"role": "user"
}
] | {
"ground_truth": "2^{99}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 0,
"problem": "One hundred concentric circles are labelled $C_{1}, C_{2}, C_{3}, \\ldots, C_{100}$. Each circle $C_{n}$ is inscribed within an equilateral triangle whose vertices are points on $C_{n+1}$. Given $C_{1}$ has a radius of $1$, what is the radius of $C_{100}$ ?"
} |
[
{
"content": "An infinite geometric sequence with common ratio $r$ sums to $91$. A new sequence starting with the same term has common ratio $r^{3}$. The sum of the new sequence produced is $81$. What was the common ratio of the original sequence?",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{9}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 1,
"problem": "An infinite geometric sequence with common ratio $r$ sums to $91$. A new sequence starting with the same term has common ratio $r^{3}$. The sum of the new sequence produced is $81$. What was the common ratio of the original sequence?"
} |
[
{
"content": "Let $A, B, C, D$, and $E$ be five equally spaced points on a line in that order. Let $F, G, H$, and $I$ all be on the same side of line $A E$ such that triangles $A F B, B G C, C H D$, and $D I E$ are equilateral with side length $1$. Let $S$ be the region consisting of the interiors of all four t... | {
"ground_truth": "\\frac{\\sqrt{13}}{2}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 2,
"problem": "Let $A, B, C, D$, and $E$ be five equally spaced points on a line in that order. Let $F, G, H$, and $I$ all be on the same side of line $A E$ such that triangles $A F B, B G C, C H D$, and $D I E$ are equilateral with side length $1$. Let $S$ be the region consisting of the interiors of al... |
[
{
"content": "If $5 f(x)-x f\\left(\\frac{1}{x}\\right)=\\frac{1}{17} x^{2}$, determine $f(3)$.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{9}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 3,
"problem": "If $5 f(x)-x f\\left(\\frac{1}{x}\\right)=\\frac{1}{17} x^{2}$, determine $f(3)$."
} |
[
{
"content": "How many ways are there to arrange $1,2,3,4,5,6$ such that no two consecutive numbers have the same remainder when divided by $3$ ?",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 4,
"problem": "How many ways are there to arrange $1,2,3,4,5,6$ such that no two consecutive numbers have the same remainder when divided by $3$ ?"
} |
[
{
"content": "Joshua is playing with his number cards. He has $9$ cards of $9$ lined up in a row. He puts a multiplication sign between two of the $9 \\mathrm{~s}$ and calculates the product of the two strings of $9 \\mathrm{~s}$. For example, one possible result is $999 \\times 999999=998999001$. Let $S$ be th... | {
"ground_truth": "72",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 5,
"problem": "Joshua is playing with his number cards. He has $9$ cards of $9$ lined up in a row. He puts a multiplication sign between two of the $9 \\mathrm{~s}$ and calculates the product of the two strings of $9 \\mathrm{~s}$. For example, one possible result is $999 \\times 999999=998999001$. Let $... |
[
{
"content": "Bruno the Bear is tasked to organize $16$ identical brown balls into $7$ bins labeled 1-7. He must distribute the balls among the bins so that each odd-labeled bin contains an odd number of balls, and each even-labeled bin contains an even number of balls (with $0$ considered even). In how many wa... | {
"ground_truth": "924",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 6,
"problem": "Bruno the Bear is tasked to organize $16$ identical brown balls into $7$ bins labeled 1-7. He must distribute the balls among the bins so that each odd-labeled bin contains an odd number of balls, and each even-labeled bin contains an even number of balls (with $0$ considered even). In how... |
[
{
"content": "Let $f(n)$ be the number obtained by increasing every prime factor in $f$ by one. For instance, $f(12)=(2+1)^{2}(3+1)=36$. What is the lowest $n$ such that $6^{2025}$ divides $f^{(n)}(2025)$, where $f^{(n)}$ denotes the $n$th iteration of $f$ ?",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 7,
"problem": "Let $f(n)$ be the number obtained by increasing every prime factor in $f$ by one. For instance, $f(12)=(2+1)^{2}(3+1)=36$. What is the lowest $n$ such that $6^{2025}$ divides $f^{(n)}(2025)$, where $f^{(n)}$ denotes the $n$th iteration of $f$ ?"
} |
[
{
"content": "How many positive integer divisors of $63^{10}$ do not end in a $1$ ?",
"role": "user"
}
] | {
"ground_truth": "173",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 8,
"problem": "How many positive integer divisors of $63^{10}$ do not end in a $1$ ?"
} |
[
{
"content": "Bruno is throwing a party and invites $n$ guests. Each pair of party guests are either friends or enemies. Each guest has exactly $12$ enemies. All guests believe the following: the friend of an enemy is an enemy. Calculate the sum of all possible values of $n$. (Please note: Bruno is not a guest ... | {
"ground_truth": "100",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 9,
"problem": "Bruno is throwing a party and invites $n$ guests. Each pair of party guests are either friends or enemies. Each guest has exactly $12$ enemies. All guests believe the following: the friend of an enemy is an enemy. Calculate the sum of all possible values of $n$. (Please note: Bruno is not ... |
[
{
"content": "In acute $\\triangle A B C$, let $D$ be the foot of the altitude from $A$ to $B C$ and $O$ be the circumcenter. Suppose that the area of $\\triangle A B D$ is equal to the area of $\\triangle A O C$. Given that $O D=2$ and $B D=3$, compute $A D$.",
"role": "user"
}
] | {
"ground_truth": "3+2\\sqrt{2}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 10,
"problem": "In acute $\\triangle A B C$, let $D$ be the foot of the altitude from $A$ to $B C$ and $O$ be the circumcenter. Suppose that the area of $\\triangle A B D$ is equal to the area of $\\triangle A O C$. Given that $O D=2$ and $B D=3$, compute $A D$."
} |
[
{
"content": "Alice has $10$ gifts $g_{1}, g_{2}, \\ldots, g_{10}$ and $10$ friends $f_{1}, f_{2}, \\ldots, f_{10}$. Gift $g_{i}$ can be given to friend $f_{j}$ if\n\n$$\ni-j=-1,0, \\text { or } 1 \\quad(\\bmod 10)\n$$\n\nHow many ways are there for Alice to pair the $10$ gifts with the $10$ friends such that e... | {
"ground_truth": "125",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 11,
"problem": "Alice has $10$ gifts $g_{1}, g_{2}, \\ldots, g_{10}$ and $10$ friends $f_{1}, f_{2}, \\ldots, f_{10}$. Gift $g_{i}$ can be given to friend $f_{j}$ if\n\n$$\ni-j=-1,0, \\text { or } 1 \\quad(\\bmod 10)\n$$\n\nHow many ways are there for Alice to pair the $10$ gifts with the $10$ friends su... |
[
{
"content": "Let $\\triangle A B C$ be an equilateral triangle with side length $1$. A real number $d$ is selected uniformly at random from the open interval $(0,0.5)$. Points $E$ and $F$ lie on sides $A C$ and $A B$, respectively, such that $A E=d$ and $A F=1-d$. Let $D$ be the intersection of lines $B E$ and... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 12,
"problem": "Let $\\triangle A B C$ be an equilateral triangle with side length $1$. A real number $d$ is selected uniformly at random from the open interval $(0,0.5)$. Points $E$ and $F$ lie on sides $A C$ and $A B$, respectively, such that $A E=d$ and $A F=1-d$. Let $D$ be the intersection of lines ... |
[
{
"content": "Define sequence $\\left\\{a_{n}\\right\\}_{n=1}^{\\infty}$ such that $a_{1}=\\frac{\\pi}{3}$ and $a_{n+1}=\\cot ^{-1}\\left(\\csc \\left(a_{n}\\right)\\right)$ for all positive integers $n$. Find the value of\n\n$$\n\\frac{1}{\\cos \\left(a_{1}\\right) \\cos \\left(a_{2}\\right) \\cos \\left(a_{3}... | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 13,
"problem": "Define sequence $\\left\\{a_{n}\\right\\}_{n=1}^{\\infty}$ such that $a_{1}=\\frac{\\pi}{3}$ and $a_{n+1}=\\cot ^{-1}\\left(\\csc \\left(a_{n}\\right)\\right)$ for all positive integers $n$. Find the value of\n\n$$\n\\frac{1}{\\cos \\left(a_{1}\\right) \\cos \\left(a_{2}\\right) \\cos \\l... |
[
{
"content": "Define $\\{x\\}$ to be the fractional part of $x$. For example, $\\{20.25\\}=0.25$ and $\\{\\pi\\}=\\pi-3$. Let $A=\\sum_{a=1}^{96} \\sum_{n=1}^{96}\\left\\{\\frac{a^{n}}{97}\\right\\}$, where $\\{x\\}$ denotes the fractional part of $x$. Compute $A$ rounded to the nearest integer.",
"role": "... | {
"ground_truth": "4529",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 14,
"problem": "Define $\\{x\\}$ to be the fractional part of $x$. For example, $\\{20.25\\}=0.25$ and $\\{\\pi\\}=\\pi-3$. Let $A=\\sum_{a=1}^{96} \\sum_{n=1}^{96}\\left\\{\\frac{a^{n}}{97}\\right\\}$, where $\\{x\\}$ denotes the fractional part of $x$. Compute $A$ rounded to the nearest integer."
} |
[
{
"content": "Find the smallest positive integer $n$ such that $n$ is divisible by exactly $25$ different positive integers.",
"role": "user"
}
] | {
"ground_truth": "1296",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 15,
"problem": "Find the smallest positive integer $n$ such that $n$ is divisible by exactly $25$ different positive integers."
} |
[
{
"content": "Two squares, $A B C D$ and $A E F G$, have equal side length $x$. They intersect at $A$ and $O$. Given that $C O=2$ and $O A=2 \\sqrt{2}$, what is $x$ ?",
"role": "user"
}
] | {
"ground_truth": "1+\\sqrt{3}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 16,
"problem": "Two squares, $A B C D$ and $A E F G$, have equal side length $x$. They intersect at $A$ and $O$. Given that $C O=2$ and $O A=2 \\sqrt{2}$, what is $x$ ?"
} |
[
{
"content": "Bruno and Brutus are running on a circular track with a $20$ foot radius. Bruno completes $5$ laps every hour, while Brutus completes $7$ laps every hour. If they start at the same point but run in opposite directions, how far along the track's circumference (in feet) from the starting point are t... | {
"ground_truth": "20\\pi",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 17,
"problem": "Bruno and Brutus are running on a circular track with a $20$ foot radius. Bruno completes $5$ laps every hour, while Brutus completes $7$ laps every hour. If they start at the same point but run in opposite directions, how far along the track's circumference (in feet) from the starting po... |
[
{
"content": "What is the smallest positive integer $n$ such that $z^{n}-1$ and $(z-\\sqrt{3})^{n}-1$ share a common complex root?",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 18,
"problem": "What is the smallest positive integer $n$ such that $z^{n}-1$ and $(z-\\sqrt{3})^{n}-1$ share a common complex root?"
} |
[
{
"content": "Consider a pond with lily pads numbered from $1$ to $12$ arranged in a circle. Bruno the frog starts on lily pad 1. Each turn, Bruno has an equal probability of making one of three moves: jumping $4$ lily pads clockwise, jumping $2$ lily pads clockwise, or jumping $1$ lily pad counterclockwise. Wh... | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 19,
"problem": "Consider a pond with lily pads numbered from $1$ to $12$ arranged in a circle. Bruno the frog starts on lily pad 1. Each turn, Bruno has an equal probability of making one of three moves: jumping $4$ lily pads clockwise, jumping $2$ lily pads clockwise, or jumping $1$ lily pad countercloc... |
[
{
"content": "$4$ bears - Aruno, Bruno, Cruno and Druno - are each given a card with a positive integer and are told that the sum of their $4$ numbers is $17$. They cannot show each other their cards, but discuss a series of observations in the following order:\n\nAruno: \"I think it is possible that the other ... | {
"ground_truth": "160",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 20,
"problem": "$4$ bears - Aruno, Bruno, Cruno and Druno - are each given a card with a positive integer and are told that the sum of their $4$ numbers is $17$. They cannot show each other their cards, but discuss a series of observations in the following order:\n\nAruno: \"I think it is possible that t... |
[
{
"content": "Digits $1$ through $9$ are placed on a $3 x 3$ square such that all rows and columns sum to the same value. Please note that diagonals do not need to sum to the same value. How many ways can this be done?",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 21,
"problem": "Digits $1$ through $9$ are placed on a $3 x 3$ square such that all rows and columns sum to the same value. Please note that diagonals do not need to sum to the same value. How many ways can this be done?"
} |
[
{
"content": "Define the operation $\\oplus$ by\n\n$$\nx \\oplus y=x y-2 x-2 y+6 .\n$$\n\nCompute all complex numbers $a$ such that\n\n$$\na \\oplus(a \\oplus(a \\oplus a))=a .\n$$",
"role": "user"
}
] | {
"ground_truth": "2,3,\\frac{3+i\\sqrt{3}}{2},\\frac{3-i\\sqrt{3}}{2}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 22,
"problem": "Define the operation $\\oplus$ by\n\n$$\nx \\oplus y=x y-2 x-2 y+6 .\n$$\n\nCompute all complex numbers $a$ such that\n\n$$\na \\oplus(a \\oplus(a \\oplus a))=a .\n$$"
} |
[
{
"content": "Define the function $f$ on positive integers\n\n$$\nf(n)= \\begin{cases}\\frac{n}{2} & \\text { if } n \\text { is even } \\\\ n+1 & \\text { if } n \\text { is odd }\\end{cases}\n$$\n\nLet $S(n)$ equal the smallest positive integer $k$ such that $f^{k}(n)=1$. How many positive integers satisfy $S... | {
"ground_truth": "89",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 23,
"problem": "Define the function $f$ on positive integers\n\n$$\nf(n)= \\begin{cases}\\frac{n}{2} & \\text { if } n \\text { is even } \\\\ n+1 & \\text { if } n \\text { is odd }\\end{cases}\n$$\n\nLet $S(n)$ equal the smallest positive integer $k$ such that $f^{k}(n)=1$. How many positive integers s... |
[
{
"content": "Let $A B C D E F$ be a convex cyclic hexagon. Suppose that $A B=D E=\\sqrt{5}, B C=E F=3$, and $C D=F A=\\sqrt{20}$. Compute the circumradius of $A B C D E F$.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1+\\sqrt{31}}{2}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 24,
"problem": "Let $A B C D E F$ be a convex cyclic hexagon. Suppose that $A B=D E=\\sqrt{5}, B C=E F=3$, and $C D=F A=\\sqrt{20}$. Compute the circumradius of $A B C D E F$."
} |
[
{
"content": "A repetend is the infinitely repeated digit sequence of a repeating decimal. What are the last three digits of the repetend of the decimal representation of $\\frac{1}{727}$, given that the repetend has a length of $726$ ? Express the answer as a three-digit number. Include preceding zeros if ther... | {
"ground_truth": "337",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 25,
"problem": "A repetend is the infinitely repeated digit sequence of a repeating decimal. What are the last three digits of the repetend of the decimal representation of $\\frac{1}{727}$, given that the repetend has a length of $726$ ? Express the answer as a three-digit number. Include preceding zero... |
[
{
"content": "Consider a $54$-deck of cards, i.e. a standard $52$-card deck together with two jokers. Ada draws cards from the deck until Ada has drawn an ace, a king, and a queen. How many cards does Ada pick up on average?",
"role": "user"
}
] | {
"ground_truth": "\\frac{737}{39}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 26,
"problem": "Consider a $54$-deck of cards, i.e. a standard $52$-card deck together with two jokers. Ada draws cards from the deck until Ada has drawn an ace, a king, and a queen. How many cards does Ada pick up on average?"
} |
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