prompt listlengths 1 1 | reward_model dict | data_source stringclasses 7
values | ability stringclasses 1
value | extra_info dict |
|---|---|---|---|---|
[
{
"content": "Let $\\omega$ be a circle, and let a line $\\ell$ intersect $\\omega$ at two points, $P$ and $Q$. Circles $\\omega_{1}$ and $\\omega_{2}$ are internally tangent to $\\omega$ at points $X$ and $Y$, respectively, and both are tangent to $\\ell$ at a common point $D$. Similarly, circles $\\omega_{3}$... | {
"ground_truth": "5\\sqrt{2}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 27,
"problem": "Let $\\omega$ be a circle, and let a line $\\ell$ intersect $\\omega$ at two points, $P$ and $Q$. Circles $\\omega_{1}$ and $\\omega_{2}$ are internally tangent to $\\omega$ at points $X$ and $Y$, respectively, and both are tangent to $\\ell$ at a common point $D$. Similarly, circles $\\o... |
[
{
"content": "Let $f$ be a degree $7$ polynomial satisfying\n$$\nf(k)=\\frac{1}{k^{2}}\n$$\n\nfor $k \\in\\{1 \\cdot 2,2 \\cdot 3, \\ldots, 8 \\cdot 9\\}$. Find $f(90)-\\frac{1}{90^{2}}$.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2431}{50}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 28,
"problem": "Let $f$ be a degree $7$ polynomial satisfying\n$$\nf(k)=\\frac{1}{k^{2}}\n$$\n\nfor $k \\in\\{1 \\cdot 2,2 \\cdot 3, \\ldots, 8 \\cdot 9\\}$. Find $f(90)-\\frac{1}{90^{2}}$."
} |
[
{
"content": "Let $\\triangle A B C$ be an isosceles triangle with $A B=A C$. Let $D$ be a point on the circumcircle of $\\triangle A B C$ on minor arc $A B$. Let $\\overline{A D}$ intersect the extension of $\\overline{B C}$ at $E$. Let $F$ be the midpoint of segment $A C$, and let $G$ be the intersection of $... | {
"ground_truth": "\\frac{9\\sqrt{30}}{4}",
"style": "rule-lighteval/MATH_v2"
} | brumo25 | MATH | {
"index": 29,
"problem": "Let $\\triangle A B C$ be an isosceles triangle with $A B=A C$. Let $D$ be a point on the circumcircle of $\\triangle A B C$ on minor arc $A B$. Let $\\overline{A D}$ intersect the extension of $\\overline{B C}$ at $E$. Let $F$ be the midpoint of segment $A C$, and let $G$ be the intersec... |
[
{
"content": "Compute the sum of the positive divisors (including $1$) of $9!$ that have units digit $1$.",
"role": "user"
}
] | {
"ground_truth": "103",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 0,
"problem": "Compute the sum of the positive divisors (including $1$) of $9!$ that have units digit $1$."
} |
[
{
"content": "Mark writes the expression $\\sqrt{\\underline{a b c d}}$ on the board, where $\\underline{a b c d}$ is a four-digit number and $a \\neq 0$. Derek, a toddler, decides to move the $a$, changing Mark's expression to $a \\sqrt{\\underline{b c d}}$. Surprisingly, these two expressions are equal. Compu... | {
"ground_truth": "3375",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 1,
"problem": "Mark writes the expression $\\sqrt{\\underline{a b c d}}$ on the board, where $\\underline{a b c d}$ is a four-digit number and $a \\neq 0$. Derek, a toddler, decides to move the $a$, changing Mark's expression to $a \\sqrt{\\underline{b c d}}$. Surprisingly, these two expressions are equa... |
[
{
"content": "Given that $x, y$, and $z$ are positive real numbers such that\n\n$$\nx^{\\log _{2}(y z)}=2^{8} \\cdot 3^{4}, \\quad y^{\\log _{2}(z x)}=2^{9} \\cdot 3^{6}, \\quad \\text { and } \\quad z^{\\log _{2}(x y)}=2^{5} \\cdot 3^{10}\n$$\ncompute the smallest possible value of $x y z$.",
"role": "user... | {
"ground_truth": "\\frac{1}{576}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 2,
"problem": "Given that $x, y$, and $z$ are positive real numbers such that\n\n$$\nx^{\\log _{2}(y z)}=2^{8} \\cdot 3^{4}, \\quad y^{\\log _{2}(z x)}=2^{9} \\cdot 3^{6}, \\quad \\text { and } \\quad z^{\\log _{2}(x y)}=2^{5} \\cdot 3^{10}\n$$\ncompute the smallest possible value of $x y z$."
} |
[
{
"content": "Let $\\lfloor z\\rfloor$ denote the greatest integer less than or equal to $z$. Compute\n\n$$\n\\sum_{j=-1000}^{1000}\\left\\lfloor\\frac{2025}{j+0.5}\\right\\rfloor\n$$",
"role": "user"
}
] | {
"ground_truth": "-984",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 3,
"problem": "Let $\\lfloor z\\rfloor$ denote the greatest integer less than or equal to $z$. Compute\n\n$$\n\\sum_{j=-1000}^{1000}\\left\\lfloor\\frac{2025}{j+0.5}\\right\\rfloor\n$$"
} |
[
{
"content": "Let $\\mathcal{S}$ be the set of all nonconstant monic polynomials $P$ with integer coefficients satisfying $P(\\sqrt{3}+\\sqrt{2})=$ $P(\\sqrt{3}-\\sqrt{2})$. If $Q$ is an element of $\\mathcal{S}$ with minimal degree, compute the only possible value of $Q(10)-Q(0)$.",
"role": "user"
}
] | {
"ground_truth": "890",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 4,
"problem": "Let $\\mathcal{S}$ be the set of all nonconstant monic polynomials $P$ with integer coefficients satisfying $P(\\sqrt{3}+\\sqrt{2})=$ $P(\\sqrt{3}-\\sqrt{2})$. If $Q$ is an element of $\\mathcal{S}$ with minimal degree, compute the only possible value of $Q(10)-Q(0)$."
} |
[
{
"content": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is prime.)",
"role": "user"
}
] | {
"ground_truth": "\\frac{1311}{2017}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 5,
"problem": "Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\\frac{r}{2025!}$. (Note that $2017$ is prime.)"
} |
[
{
"content": "There exists a unique triple $(a, b, c)$ of positive real numbers that satisfies the equations\n$$\n2\\left(a^{2}+1\\right)=3\\left(b^{2}+1\\right)=4\\left(c^{2}+1\\right) \\quad \\text { and } \\quad a b+b c+c a=1\n$$\n\nCompute $a+b+c$.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9 \\sqrt{23}}{23}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 6,
"problem": "There exists a unique triple $(a, b, c)$ of positive real numbers that satisfies the equations\n$$\n2\\left(a^{2}+1\\right)=3\\left(b^{2}+1\\right)=4\\left(c^{2}+1\\right) \\quad \\text { and } \\quad a b+b c+c a=1\n$$\n\nCompute $a+b+c$."
} |
[
{
"content": "Define $\\operatorname{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0$. Compute\n\n$$\n\\sum_{n=1}^{\\infty} \\frac{\\operatorname{sgn}\\left(\\sin \\left(2^{n}\\right)\\right)}{2^{n}}\n$$\n(The arguments to sin are in radians.)",
"role": "user"
}
... | {
"ground_truth": "1-\\frac{2}{\\pi}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 7,
"problem": "Define $\\operatorname{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0$. Compute\n\n$$\n\\sum_{n=1}^{\\infty} \\frac{\\operatorname{sgn}\\left(\\sin \\left(2^{n}\\right)\\right)}{2^{n}}\n$$\n(The arguments to sin are in radians.)"
} |
[
{
"content": "Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \\in\\{1,2,3, \\ldots, 2027\\}$,\n$$\nf(n)= \\begin{cases}1 & \\text { if } n \\text { is a perfect square } \\\\ 0 & \\text { otherwise }\\end{cases}\n$$\n\nSuppose that $\\frac{a}{b}$ is the coefficient of $x^{2025}$ ... | {
"ground_truth": "1037",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 8,
"problem": "Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \\in\\{1,2,3, \\ldots, 2027\\}$,\n$$\nf(n)= \\begin{cases}1 & \\text { if } n \\text { is a perfect square } \\\\ 0 & \\text { otherwise }\\end{cases}\n$$\n\nSuppose that $\\frac{a}{b}$ is the coefficient of $x^... |
[
{
"content": "Let $a, b$, and $c$ be pairwise distinct complex numbers such that\n$$\na^{2}=b+6, \\quad b^{2}=c+6, \\quad \\text { and } \\quad c^{2}=a+6\n$$\n\nCompute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \\LaTeX expressions.",
"role... | {
"ground_truth": "\\frac{-1+\\sqrt{17}}{2}, \\frac{-1-\\sqrt{17}}{2}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 9,
"problem": "Let $a, b$, and $c$ be pairwise distinct complex numbers such that\n$$\na^{2}=b+6, \\quad b^{2}=c+6, \\quad \\text { and } \\quad c^{2}=a+6\n$$\n\nCompute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \\LaTeX expressions."
} |
[
{
"content": "Compute the number of ways to arrange the numbers $1,2,3,4,5,6$, and $7$ around a circle such that the product of every pair of adjacent numbers on the circle is at most 20. (Rotations and reflections count as different arrangements.)",
"role": "user"
}
] | {
"ground_truth": "56",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 10,
"problem": "Compute the number of ways to arrange the numbers $1,2,3,4,5,6$, and $7$ around a circle such that the product of every pair of adjacent numbers on the circle is at most 20. (Rotations and reflections count as different arrangements.)"
} |
[
{
"content": "Kevin the frog in on the bottom-left lily pad of a $3 \\times 3$ grid of lily pads, and his home is at the topright lily pad. He can only jump between two lily pads which are horizontally or vertically adjacent. Compute the number of ways to remove $4$ of the lily pads so that the bottom-left and ... | {
"ground_truth": "29",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 11,
"problem": "Kevin the frog in on the bottom-left lily pad of a $3 \\times 3$ grid of lily pads, and his home is at the topright lily pad. He can only jump between two lily pads which are horizontally or vertically adjacent. Compute the number of ways to remove $4$ of the lily pads so that the bottom-... |
[
{
"content": "Ben has $16$ balls labeled $1,2,3, \\ldots, 16$, as well as $4$ indistinguishable boxes. Two balls are \\emph{neighbors} if their labels differ by $1$. Compute the number of ways for him to put $4$ balls in each box such that each ball is in the same box as at least one of its neighbors. (The orde... | {
"ground_truth": "105",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 12,
"problem": "Ben has $16$ balls labeled $1,2,3, \\ldots, 16$, as well as $4$ indistinguishable boxes. Two balls are \\emph{neighbors} if their labels differ by $1$. Compute the number of ways for him to put $4$ balls in each box such that each ball is in the same box as at least one of its neighbors. ... |
[
{
"content": "Sophie is at $(0,0)$ on a coordinate grid and would like to get to $(3,3)$. If Sophie is at $(x, y)$, in a single step she can move to one of $(x+1, y),(x, y+1),(x-1, y+1)$, or $(x+1, y-1)$. She cannot revisit any points along her path, and neither her $x$-coordinate nor her $y$-coordinate can eve... | {
"ground_truth": "2304",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 13,
"problem": "Sophie is at $(0,0)$ on a coordinate grid and would like to get to $(3,3)$. If Sophie is at $(x, y)$, in a single step she can move to one of $(x+1, y),(x, y+1),(x-1, y+1)$, or $(x+1, y-1)$. She cannot revisit any points along her path, and neither her $x$-coordinate nor her $y$-coordinat... |
[
{
"content": "In an $11 \\times 11$ grid of cells, each pair of edge-adjacent cells is connected by a door. Karthik wants to walk a path in this grid. He can start in any cell, but he must end in the same cell he started in, and he cannot go through any door more than once (not even in opposite directions). Com... | {
"ground_truth": "200",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 14,
"problem": "In an $11 \\times 11$ grid of cells, each pair of edge-adjacent cells is connected by a door. Karthik wants to walk a path in this grid. He can start in any cell, but he must end in the same cell he started in, and he cannot go through any door more than once (not even in opposite directi... |
[
{
"content": "Compute the number of ways to pick two rectangles in a $5 \\times 5$ grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are chosen does not matter.",
"rol... | {
"ground_truth": "6300",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 15,
"problem": "Compute the number of ways to pick two rectangles in a $5 \\times 5$ grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are chosen does not matter."
... |
[
{
"content": "Compute the number of ways to arrange $3$ copies of each of the $26$ lowercase letters of the English alphabet such that for any two distinct letters $x_{1}$ and $x_{2}$, the number of $x_{2}$ 's between the first and second occurrences of $x_{1}$ equals the number of $x_{2}$ 's between the second... | {
"ground_truth": "2^{25} \\cdot 26!",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 16,
"problem": "Compute the number of ways to arrange $3$ copies of each of the $26$ lowercase letters of the English alphabet such that for any two distinct letters $x_{1}$ and $x_{2}$, the number of $x_{2}$ 's between the first and second occurrences of $x_{1}$ equals the number of $x_{2}$ 's between t... |
[
{
"content": "Albert writes $2025$ numbers $a_{1}, \\ldots, a_{2025}$ in a circle on a blackboard. Initially, each of the numbers is uniformly and independently sampled at random from the interval $[0,1]$. Then, each second, he \\emph{simultaneously} replaces $a_{i}$ with $\\max \\left(a_{i-1}, a_{i}, a_{i+1}\\... | {
"ground_truth": "\\frac{2025}{101}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 17,
"problem": "Albert writes $2025$ numbers $a_{1}, \\ldots, a_{2025}$ in a circle on a blackboard. Initially, each of the numbers is uniformly and independently sampled at random from the interval $[0,1]$. Then, each second, he \\emph{simultaneously} replaces $a_{i}$ with $\\max \\left(a_{i-1}, a_{i}, ... |
[
{
"content": "Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{9}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 18,
"problem": "Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon."
} |
[
{
"content": "The circumference of a circle is divided into $45$ arcs, each of length $1$. Initially, there are $15$ snakes, each of length $1$, occupying every third arc. Every second, each snake independently moves either one arc left or one arc right, each with probability $\\frac{1}{2}$. If two snakes ever ... | {
"ground_truth": "\\frac{448}{3}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 19,
"problem": "The circumference of a circle is divided into $45$ arcs, each of length $1$. Initially, there are $15$ snakes, each of length $1$, occupying every third arc. Every second, each snake independently moves either one arc left or one arc right, each with probability $\\frac{1}{2}$. If two sna... |
[
{
"content": "Equilateral triangles $\\triangle A B C$ and $\\triangle D E F$ are drawn such that points $B, E, F$, and $C$ lie on a line in this order, and point $D$ lies inside triangle $\\triangle A B C$. If $B E=14, E F=15$, and $F C=16$, compute $A D$.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 20,
"problem": "Equilateral triangles $\\triangle A B C$ and $\\triangle D E F$ are drawn such that points $B, E, F$, and $C$ lie on a line in this order, and point $D$ lies inside triangle $\\triangle A B C$. If $B E=14, E F=15$, and $F C=16$, compute $A D$."
} |
[
{
"content": "In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths $16$ and $36$ hang perpendicularly from the ceiling, while two stalagmites of heights $25$ and $49$ grow perpendicularly from the ground. If the tips of these four structures form the vertices of a square in so... | {
"ground_truth": "63",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 21,
"problem": "In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths $16$ and $36$ hang perpendicularly from the ceiling, while two stalagmites of heights $25$ and $49$ grow perpendicularly from the ground. If the tips of these four structures form the vertices of a squ... |
[
{
"content": "Point $P$ lies inside square $A B C D$ such that the areas of $\\triangle P A B, \\triangle P B C, \\triangle P C D$, and $\\triangle P D A$ are 1, $2,3$, and $4$, in some order. Compute $P A \\cdot P B \\cdot P C \\cdot P D$.",
"role": "user"
}
] | {
"ground_truth": "8\\sqrt{10}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 22,
"problem": "Point $P$ lies inside square $A B C D$ such that the areas of $\\triangle P A B, \\triangle P B C, \\triangle P C D$, and $\\triangle P D A$ are 1, $2,3$, and $4$, in some order. Compute $P A \\cdot P B \\cdot P C \\cdot P D$."
} |
[
{
"content": "A semicircle is inscribed in another semicircle if the smaller semicircle's diameter is a chord of the larger semicircle, and the smaller semicircle's arc is tangent to the diameter of the larger semicircle.\nSemicircle $S_{1}$ is inscribed in a semicircle $S_{2}$, which is inscribed in another se... | {
"ground_truth": "20",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 23,
"problem": "A semicircle is inscribed in another semicircle if the smaller semicircle's diameter is a chord of the larger semicircle, and the smaller semicircle's arc is tangent to the diameter of the larger semicircle.\nSemicircle $S_{1}$ is inscribed in a semicircle $S_{2}$, which is inscribed in a... |
[
{
"content": "Let $\\triangle A B C$ be an equilateral triangle with side length $6$. Let $P$ be a point inside triangle $\\triangle A B C$ such that $\\angle B P C=120^{\\circ}$. The circle with diameter $\\overline{A P}$ meets the circumcircle of $\\triangle A B C$ again at $X \\neq A$. Given that $A X=5$, co... | {
"ground_truth": "\\sqrt{23}-2 \\sqrt{3}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 24,
"problem": "Let $\\triangle A B C$ be an equilateral triangle with side length $6$. Let $P$ be a point inside triangle $\\triangle A B C$ such that $\\angle B P C=120^{\\circ}$. The circle with diameter $\\overline{A P}$ meets the circumcircle of $\\triangle A B C$ again at $X \\neq A$. Given that $A... |
[
{
"content": "Trapezoid $A B C D$, with $A B \\| C D$, has side lengths $A B=11, B C=8, C D=19$, and $D A=4$. Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\\triangle A B C, \\triangle B C D$, $\\triangle C D A$, and $\\triangle D A B$.",
"role": "user"
}
] | {
"ground_truth": "9\\sqrt{15}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 25,
"problem": "Trapezoid $A B C D$, with $A B \\| C D$, has side lengths $A B=11, B C=8, C D=19$, and $D A=4$. Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\\triangle A B C, \\triangle B C D$, $\\triangle C D A$, and $\\triangle D A B$."
} |
[
{
"content": "Point $P$ is inside triangle $\\triangle A B C$ such that $\\angle A B P=\\angle A C P$. Given that $A B=6, A C=8, B C=7$, and $\\frac{B P}{P C}=\\frac{1}{2}$, compute $\\frac{[B P C]}{[A B C]}$.\n(Here, $[X Y Z]$ denotes the area of $\\triangle X Y Z$ ).",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{18}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 26,
"problem": "Point $P$ is inside triangle $\\triangle A B C$ such that $\\angle A B P=\\angle A C P$. Given that $A B=6, A C=8, B C=7$, and $\\frac{B P}{P C}=\\frac{1}{2}$, compute $\\frac{[B P C]}{[A B C]}$.\n(Here, $[X Y Z]$ denotes the area of $\\triangle X Y Z$ )."
} |
[
{
"content": "Let $A B C D$ be an isosceles trapezoid such that $C D>A B=4$. Let $E$ be a point on line $C D$ such that $D E=2$ and $D$ lies between $E$ and $C$. Let $M$ be the midpoint of $\\overline{A E}$. Given that points $A, B, C, D$, and $M$ lie on a circle with radius $5$, compute $M D$.",
"role": "u... | {
"ground_truth": "\\sqrt{6}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 27,
"problem": "Let $A B C D$ be an isosceles trapezoid such that $C D>A B=4$. Let $E$ be a point on line $C D$ such that $D E=2$ and $D$ lies between $E$ and $C$. Let $M$ be the midpoint of $\\overline{A E}$. Given that points $A, B, C, D$, and $M$ lie on a circle with radius $5$, compute $M D$."
} |
[
{
"content": "Let $A B C D$ be a rectangle with $B C=24$. Point $X$ lies inside the rectangle such that $\\angle A X B=90^{\\circ}$. Given that triangles $\\triangle A X D$ and $\\triangle B X C$ are both acute and have circumradii $13$ and $15$, respectively, compute $A B$.",
"role": "user"
}
] | {
"ground_truth": "14+4\\sqrt{37}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 28,
"problem": "Let $A B C D$ be a rectangle with $B C=24$. Point $X$ lies inside the rectangle such that $\\angle A X B=90^{\\circ}$. Given that triangles $\\triangle A X D$ and $\\triangle B X C$ are both acute and have circumradii $13$ and $15$, respectively, compute $A B$."
} |
[
{
"content": "A plane $\\mathcal{P}$ intersects a rectangular prism at a hexagon which has side lengths $45,66,63,55,54$, and 77, in that order. Compute the distance from the center of the rectangular prism to $\\mathcal{P}$.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{\\frac{95}{24}}",
"style": "rule-lighteval/MATH_v2"
} | hmmt25 | MATH | {
"index": 29,
"problem": "A plane $\\mathcal{P}$ intersects a rectangular prism at a hexagon which has side lengths $45,66,63,55,54$, and 77, in that order. Compute the distance from the center of the rectangular prism to $\\mathcal{P}$."
} |
[
{
"content": "Four runners are preparing to begin a $1$-mile race from the same starting line. When the race starts, runners Alice, Bob, and Charlie all travel at constant speeds of $8 \\mathrm{mph}, 4 \\mathrm{mph}$, and $2$ mph, respectively. The fourth runner, Dave, is initially half as slow as Charlie, but ... | {
"ground_truth": "\\frac{13}{32}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 0,
"problem": "Four runners are preparing to begin a $1$-mile race from the same starting line. When the race starts, runners Alice, Bob, and Charlie all travel at constant speeds of $8 \\mathrm{mph}, 4 \\mathrm{mph}$, and $2$ mph, respectively. The fourth runner, Dave, is initially half as slow as Charl... |
[
{
"content": "I plotted the graphs $y=(x-0)^{2}, y=(x-5)^{2}, \\ldots y=(x-45)^{2}$. I also draw a line $y=k$, and notice that it intersects the set of parabolas at $19$ distinct points. What is $k$ ?",
"role": "user"
}
] | {
"ground_truth": "\\frac{2025}{4}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 1,
"problem": "I plotted the graphs $y=(x-0)^{2}, y=(x-5)^{2}, \\ldots y=(x-45)^{2}$. I also draw a line $y=k$, and notice that it intersects the set of parabolas at $19$ distinct points. What is $k$ ?"
} |
[
{
"content": "Compute $3^{3^{\\cdots^{3}}} \\bmod 333$, where there are $3^{3^{3}} 3$ 's in the exponent.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 2,
"problem": "Compute $3^{3^{\\cdots^{3}}} \\bmod 333$, where there are $3^{3^{3}} 3$ 's in the exponent."
} |
[
{
"content": "Consider the system of equations\n\n$$\n\\begin{gathered}\n\\log _{x} y+\\log _{y} z+\\log _{z} x=8 \\\\\n\\log _{\\log _{y} x} z=-3 \\\\\n\\log _{z} y+\\log _{x} z=16\n\\end{gathered}\n$$\n\nFind $z$.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 3,
"problem": "Consider the system of equations\n\n$$\n\\begin{gathered}\n\\log _{x} y+\\log _{y} z+\\log _{z} x=8 \\\\\n\\log _{\\log _{y} x} z=-3 \\\\\n\\log _{z} y+\\log _{x} z=16\n\\end{gathered}\n$$\n\nFind $z$."
} |
[
{
"content": "Consider all positive multiples of $77$ less than $1,000,000$. What is the sum of all the odd digits that show up?",
"role": "user"
}
] | {
"ground_truth": "194832",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 4,
"problem": "Consider all positive multiples of $77$ less than $1,000,000$. What is the sum of all the odd digits that show up?"
} |
[
{
"content": "Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[-1,1]$. Find the probability that\n\n$$\n|x|+|y|+1 \\leq 3 \\min \\{|x+y+1|,|x+y-1|\\} .\n$$",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{16}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 5,
"problem": "Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[-1,1]$. Find the probability that\n\n$$\n|x|+|y|+1 \\leq 3 \\min \\{|x+y+1|,|x+y-1|\\} .\n$$"
} |
[
{
"content": "Consider a recursively defined sequence $a_{n}$ with $a_{1}=1$ such that, for $n \\geq 2, a_{n}$ is formed by appending the last digit of $n$ to the end of $a_{n-1}$. For a positive integer $m$, let $\\nu_{3}(m)$ be the largest integer $t$ such that $3^{t} \\mid m$. Compute\n\n$$\n\\sum_{n=1}^{810... | {
"ground_truth": "930",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 6,
"problem": "Consider a recursively defined sequence $a_{n}$ with $a_{1}=1$ such that, for $n \\geq 2, a_{n}$ is formed by appending the last digit of $n$ to the end of $a_{n-1}$. For a positive integer $m$, let $\\nu_{3}(m)$ be the largest integer $t$ such that $3^{t} \\mid m$. Compute\n\n$$\n\\sum_{n... |
[
{
"content": "Let $P(x)=x^{4}+20 x^{3}+29 x^{2}-666 x+2025$. It is known that $P(x)>0$ for every real $x$.\n\nThere is a root $r$ for $P$ in the first quadrant of the complex plane that can be expressed as $r=\\frac{1}{2}(a+b i+\\sqrt{c+d i})$, where $a, b, c, d$ are integers. Find $a+b+c+d$.",
"role": "use... | {
"ground_truth": "322",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 7,
"problem": "Let $P(x)=x^{4}+20 x^{3}+29 x^{2}-666 x+2025$. It is known that $P(x)>0$ for every real $x$.\n\nThere is a root $r$ for $P$ in the first quadrant of the complex plane that can be expressed as $r=\\frac{1}{2}(a+b i+\\sqrt{c+d i})$, where $a, b, c, d$ are integers. Find $a+b+c+d$."
} |
[
{
"content": "Find the largest prime factor of $45^{5}-1$.",
"role": "user"
}
] | {
"ground_truth": "2851",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 8,
"problem": "Find the largest prime factor of $45^{5}-1$."
} |
[
{
"content": "Let $a_{n}$ be a recursively defined sequence with $a_{0}=2024$ and $a_{n+1}=a_{n}^{3}+5 a_{n}^{2}+10 a_{n}+6$ for $n \\geq 0$. Determine the value of\n\n$$\n\\sum_{n=0}^{\\infty} \\frac{2^{n}\\left(a_{n}+1\\right)}{a_{n}^{2}+3 a_{n}+4}\n$$",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2026}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 9,
"problem": "Let $a_{n}$ be a recursively defined sequence with $a_{0}=2024$ and $a_{n+1}=a_{n}^{3}+5 a_{n}^{2}+10 a_{n}+6$ for $n \\geq 0$. Determine the value of\n\n$$\n\\sum_{n=0}^{\\infty} \\frac{2^{n}\\left(a_{n}+1\\right)}{a_{n}^{2}+3 a_{n}+4}\n$$"
} |
[
{
"content": "Robert has five beads in his hand, with the letters C, M, I, M, and C, and he wants to make a circular bracelet spelling \"CMIMC.\" However, the power went out, so Robert can no longer see the beads in his hand. Thus, he puts the five beads on the bracelet randomly, hoping that the bracelet, when ... | {
"ground_truth": "\\frac{1}{6}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 10,
"problem": "Robert has five beads in his hand, with the letters C, M, I, M, and C, and he wants to make a circular bracelet spelling \"CMIMC.\" However, the power went out, so Robert can no longer see the beads in his hand. Thus, he puts the five beads on the bracelet randomly, hoping that the bracel... |
[
{
"content": "Every day, Pinky the flamingo eats either $1$ or $2$ shrimp, each with equal probability. Once Pinky has consumed $10$ or more shrimp in total, its skin will turn pink. Once Pinky has consumed $11$ or more shrimp in total, it will get sick. What is the probability that Pinky does not get sick on t... | {
"ground_truth": "\\frac{683}{1024}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 11,
"problem": "Every day, Pinky the flamingo eats either $1$ or $2$ shrimp, each with equal probability. Once Pinky has consumed $10$ or more shrimp in total, its skin will turn pink. Once Pinky has consumed $11$ or more shrimp in total, it will get sick. What is the probability that Pinky does not get ... |
[
{
"content": "There are $34$ friends are sitting in a circle playing the following game. Every round, four of them are chosen at random, and have a rap battle. The winner of the rap battle stays in the circle and the other three leave. This continues until one player remains. Everyone has equal rapping ability,... | {
"ground_truth": "\\frac{2}{17}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 12,
"problem": "There are $34$ friends are sitting in a circle playing the following game. Every round, four of them are chosen at random, and have a rap battle. The winner of the rap battle stays in the circle and the other three leave. This continues until one player remains. Everyone has equal rapping... |
[
{
"content": "Let $n$ and $k$ be positive integers, with $k \\leq n$. Define a (simple, undirected) graph $G_{n, k}$ as follows: its vertices are all of the binary strings of length $n$, and there is an edge between two strings if and only if they differ in exactly $k$ positions. If $c_{n, k}$ denotes the numbe... | {
"ground_truth": "1088",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 13,
"problem": "Let $n$ and $k$ be positive integers, with $k \\leq n$. Define a (simple, undirected) graph $G_{n, k}$ as follows: its vertices are all of the binary strings of length $n$, and there is an edge between two strings if and only if they differ in exactly $k$ positions. If $c_{n, k}$ denotes ... |
[
{
"content": "Consider a $12$-card deck containing all four suits of $2,3$, and $4$. A double is defined as two cards directly next to each other in the deck, with the same value. Suppose we scan the deck left to right, and whenever we encounter a double, we remove all the cards up to that point (including the ... | {
"ground_truth": "\\frac{138}{55}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 14,
"problem": "Consider a $12$-card deck containing all four suits of $2,3$, and $4$. A double is defined as two cards directly next to each other in the deck, with the same value. Suppose we scan the deck left to right, and whenever we encounter a double, we remove all the cards up to that point (inclu... |
[
{
"content": "Consider a $4 \\times 4$ grid of squares. We place coins in some of the grid squares so that no two coins are orthogonally adjacent, and each $2 \\times 2$ square in the grid has at least one coin. How many ways are there to place the coins?",
"role": "user"
}
] | {
"ground_truth": "256",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 15,
"problem": "Consider a $4 \\times 4$ grid of squares. We place coins in some of the grid squares so that no two coins are orthogonally adjacent, and each $2 \\times 2$ square in the grid has at least one coin. How many ways are there to place the coins?"
} |
[
{
"content": "Alan is bored one day and decides to write down all the divisors of $1260^{2}$ on a wall. After writing down all of them, he realizes he wrote them on the wrong wall and needs to erase all his work. Every second, he picks a random divisor which is still on the wall and instantly erases it and ever... | {
"ground_truth": "\\frac{1507^{2}}{360^{2}}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 16,
"problem": "Alan is bored one day and decides to write down all the divisors of $1260^{2}$ on a wall. After writing down all of them, he realizes he wrote them on the wrong wall and needs to erase all his work. Every second, he picks a random divisor which is still on the wall and instantly erases it... |
[
{
"content": "Divide a regular 8960-gon into non-overlapping parallelograms. Suppose that $R$ of these parallelograms are rectangles. What is the minimum possible value of $R$ ?",
"role": "user"
}
] | {
"ground_truth": "2240",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 17,
"problem": "Divide a regular 8960-gon into non-overlapping parallelograms. Suppose that $R$ of these parallelograms are rectangles. What is the minimum possible value of $R$ ?"
} |
[
{
"content": "Let $p(k)$ be the probability that if we choose a uniformly random subset $S$ of $\\{1,2, \\ldots, 18\\}$, then $|S| \\equiv k$ $(\\bmod 5)$.\nEvaluate\n\n$$\n\\sum_{k=0}^{4}\\left|p(k)-\\frac{1}{5}\\right|\n$$",
"role": "user"
}
] | {
"ground_truth": "\\frac{9349}{327680}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 18,
"problem": "Let $p(k)$ be the probability that if we choose a uniformly random subset $S$ of $\\{1,2, \\ldots, 18\\}$, then $|S| \\equiv k$ $(\\bmod 5)$.\nEvaluate\n\n$$\n\\sum_{k=0}^{4}\\left|p(k)-\\frac{1}{5}\\right|\n$$"
} |
[
{
"content": "Let $a_{n}$ be the number of ways to express $n$ as an ordered sum of powers of $3$. For example, $a_{4}=3$, since\n\n$$\n4=1+1+1+1=1+3=3+1 .\n$$\n\nLet $b_{n}$ denote the remainder upon dividing $a_{n}$ by $3$. Evaluate\n\n$$\n\\sum_{n=1}^{3^{2025}} b_{n} .\n$$",
"role": "user"
}
] | {
"ground_truth": "4102652",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 19,
"problem": "Let $a_{n}$ be the number of ways to express $n$ as an ordered sum of powers of $3$. For example, $a_{4}=3$, since\n\n$$\n4=1+1+1+1=1+3=3+1 .\n$$\n\nLet $b_{n}$ denote the remainder upon dividing $a_{n}$ by $3$. Evaluate\n\n$$\n\\sum_{n=1}^{3^{2025}} b_{n} .\n$$"
} |
[
{
"content": "I'm given a square of side length $7$, and I want to make a regular tetrahedron from it. Specifically, my strategy is to cut out a net. If I cut out a parallelogram-shaped net that yields the biggest regular tetrahedron, what is the surface area of the resulting tetrahedron?",
"role": "user"
... | {
"ground_truth": "14\\sqrt{3}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 20,
"problem": "I'm given a square of side length $7$, and I want to make a regular tetrahedron from it. Specifically, my strategy is to cut out a net. If I cut out a parallelogram-shaped net that yields the biggest regular tetrahedron, what is the surface area of the resulting tetrahedron?"
} |
[
{
"content": "Given a cube of side length $4$, place eight spheres of radius $1$ inside the cube so that each sphere is externally tangent to three others. What is the radius of the largest sphere contained inside the cube which is externally tangent to all eight?",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}-1",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 21,
"problem": "Given a cube of side length $4$, place eight spheres of radius $1$ inside the cube so that each sphere is externally tangent to three others. What is the radius of the largest sphere contained inside the cube which is externally tangent to all eight?"
} |
[
{
"content": "Let $A B$ be a segment of length $1$. Let $\\odot A, \\odot B$ be circles with radius $\\overline{A B}$ centered at $A, B$. Denote their intersection points $C, D$. Draw circles $\\odot C, \\odot D$ with radius $\\overline{C D}$. Denote the intersection points of $\\odot C$ and $\\odot D$ by $E, F... | {
"ground_truth": "\\frac{11\\sqrt{3}}{4}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 22,
"problem": "Let $A B$ be a segment of length $1$. Let $\\odot A, \\odot B$ be circles with radius $\\overline{A B}$ centered at $A, B$. Denote their intersection points $C, D$. Draw circles $\\odot C, \\odot D$ with radius $\\overline{C D}$. Denote the intersection points of $\\odot C$ and $\\odot D$... |
[
{
"content": "Let $A B C D E F$ be a regular hexagon with side length $1$, and let $G$ be the midpoint of side $\\overline{C D}$, and define $H$ to be the unique point on side $\\overline{D E}$ such that $A G H F$ is a trapezoid. Find the length of the altitude dropped from point $H$ to $\\overline{A G}$.",
... | {
"ground_truth": "\\frac{2\\sqrt{39}}{13}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 23,
"problem": "Let $A B C D E F$ be a regular hexagon with side length $1$, and let $G$ be the midpoint of side $\\overline{C D}$, and define $H$ to be the unique point on side $\\overline{D E}$ such that $A G H F$ is a trapezoid. Find the length of the altitude dropped from point $H$ to $\\overline{A G... |
[
{
"content": "Let $\\triangle A B C$ be an equilateral triangle. Let $E_{A B}$ be the ellipse with foci $A, B$ passing through $C$, and in the parallel manner define $E_{B C}, E_{A C}$. Let $\\triangle G H I$ be a (nondegenerate) triangle with vertices where two ellipses intersect such that the edges of $\\tria... | {
"ground_truth": "\\frac{12\\sqrt{3}+4}{13}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 24,
"problem": "Let $\\triangle A B C$ be an equilateral triangle. Let $E_{A B}$ be the ellipse with foci $A, B$ passing through $C$, and in the parallel manner define $E_{B C}, E_{A C}$. Let $\\triangle G H I$ be a (nondegenerate) triangle with vertices where two ellipses intersect such that the edges o... |
[
{
"content": "Points $A, B, C, D, E$, and $F$ lie on a sphere with center $O$ and radius $R$ such that $\\overline{A B}, \\overline{C D}$, and $\\overline{E F}$ are pairwise perpendicular and all meet at a point $X$ inside the sphere. If $A X=1$, $C X=\\sqrt{2}, E X=2$, and $O X=\\frac{\\sqrt{2}}{2}$, compute t... | {
"ground_truth": "\\frac{31}{7}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 25,
"problem": "Points $A, B, C, D, E$, and $F$ lie on a sphere with center $O$ and radius $R$ such that $\\overline{A B}, \\overline{C D}$, and $\\overline{E F}$ are pairwise perpendicular and all meet at a point $X$ inside the sphere. If $A X=1$, $C X=\\sqrt{2}, E X=2$, and $O X=\\frac{\\sqrt{2}}{2}$, ... |
[
{
"content": "Let $A B C$ be a triangle with altitude $\\overline{A F}$. Let $A B=5, A C=8, B C=7$. Let $P$ be on $\\overline{A F}$ such that it lies between $A$ and $F$. Let $\\omega_{1}, \\omega_{2}$ be the circumcircles of $A P B, A P C$ respectively. Let $\\overline{B C}$ intersect $\\omega_{1}$ at $B^{\\pr... | {
"ground_truth": "\\frac{7\\sqrt{3}}{3}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 26,
"problem": "Let $A B C$ be a triangle with altitude $\\overline{A F}$. Let $A B=5, A C=8, B C=7$. Let $P$ be on $\\overline{A F}$ such that it lies between $A$ and $F$. Let $\\omega_{1}, \\omega_{2}$ be the circumcircles of $A P B, A P C$ respectively. Let $\\overline{B C}$ intersect $\\omega_{1}$ at... |
[
{
"content": "Let $\\omega$ be a circle with diameter $\\overline{A B}$, center $O$, and cyclic quadrilateral $A B C D$ inscribed in it, with $C$ and $D$ on the same side of $\\overline{A B}$. Let $A B=20, B C=13, A D=7$. Let $\\overleftrightarrow{B C}$ and $\\overleftrightarrow{A D}$ intersect at $E$. Let the ... | {
"ground_truth": "\\frac{97-3\\sqrt{1001}}{2}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 27,
"problem": "Let $\\omega$ be a circle with diameter $\\overline{A B}$, center $O$, and cyclic quadrilateral $A B C D$ inscribed in it, with $C$ and $D$ on the same side of $\\overline{A B}$. Let $A B=20, B C=13, A D=7$. Let $\\overleftrightarrow{B C}$ and $\\overleftrightarrow{A D}$ intersect at $E$.... |
[
{
"content": "Define the ratio of an ellipse to be the length of the major axis divided by the length of the minor axis.\n\nGiven a trapezoid $A B C D$ with $A B \\| D C$ and that $\\angle A D C$ is a right angle, with $A B=18, A D=$ $33, C D=130$, find the smallest ratio of any ellipse that goes through all ve... | {
"ground_truth": "\\frac{11}{3}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 28,
"problem": "Define the ratio of an ellipse to be the length of the major axis divided by the length of the minor axis.\n\nGiven a trapezoid $A B C D$ with $A B \\| D C$ and that $\\angle A D C$ is a right angle, with $A B=18, A D=$ $33, C D=130$, find the smallest ratio of any ellipse that goes throu... |
[
{
"content": "Let $\\triangle A B C$ exist such that $A B=6, B C=8, A C=10$. Let $P$ lie on the circumcircle of $A B C, \\omega$, such that $P$ lies strictly on the arc in between $B$ and $C$ (i.e. $P \\neq B, C$ ). Drop altitudes from $P$ to $B C, A C$ at points $J$ and $Q$ respectively. Let $l$ be a line thro... | {
"ground_truth": "9.94",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 29,
"problem": "Let $\\triangle A B C$ exist such that $A B=6, B C=8, A C=10$. Let $P$ lie on the circumcircle of $A B C, \\omega$, such that $P$ lies strictly on the arc in between $B$ and $C$ (i.e. $P \\neq B, C$ ). Drop altitudes from $P$ to $B C, A C$ at points $J$ and $Q$ respectively. Let $l$ be a ... |
[
{
"content": "I define a \"good day\" as a day when both the day and the month evenly divide the concatenation of the two. For example, today (March 15) is a good day since $3$ and $15$ both divide 315. However, March $9$ is not a good day since $9$ does not divide 39. How many good days are in March, April, an... | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 30,
"problem": "I define a \"good day\" as a day when both the day and the month evenly divide the concatenation of the two. For example, today (March 15) is a good day since $3$ and $15$ both divide 315. However, March $9$ is not a good day since $9$ does not divide 39. How many good days are in March, ... |
[
{
"content": "We are searching for the number $7$ in the following binary tree:\n\n 4\n / \\\n 2 6\n / \\ / \\\n1 3 5 7\n\n- **Level 1 (Root):** 4\n- **Level 2:** 2 (left), 6 (right)\n- **Level 3:**\n - 2's children: 1 (left), 3 (right)\n - 6's children: 5 (left), 7 (right)\n\nWe use the fo... | {
"ground_truth": "\\frac{11}{4}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 31,
"problem": "We are searching for the number $7$ in the following binary tree:\n\n 4\n / \\\n 2 6\n / \\ / \\\n1 3 5 7\n\n- **Level 1 (Root):** 4\n- **Level 2:** 2 (left), 6 (right)\n- **Level 3:**\n - 2's children: 1 (left), 3 (right)\n - 6's children: 5 (left), 7 (right)\n\nWe u... |
[
{
"content": "Let $f(x)=x^{4}-4 x^{2}+2$. Find the smallest natural $n \\in \\mathbb{N}$ such that there exists $k, c \\in \\mathbb{N}$ with\n\n$$\n\\left|f^{k}\\left(\\frac{n^{2}+1}{n}\\right)-c^{144}\\right|<\\frac{1}{100} .\n$$",
"role": "user"
}
] | {
"ground_truth": "512",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 32,
"problem": "Let $f(x)=x^{4}-4 x^{2}+2$. Find the smallest natural $n \\in \\mathbb{N}$ such that there exists $k, c \\in \\mathbb{N}$ with\n\n$$\n\\left|f^{k}\\left(\\frac{n^{2}+1}{n}\\right)-c^{144}\\right|<\\frac{1}{100} .\n$$"
} |
[
{
"content": "A non-self intersecting hexagon $R A N D O M$ is formed by assigning the labels $R, A, N, D, O, M$ in some order to the points\n$$\n(0,0),(10,0),(10,10),(0,10),(3,4),(6,2) .\n$$\n\nLet $a_{\\text {max }}$ be the greatest possible area of $R A N D O M$ and $a_{\\text {min }}$ the least possible are... | {
"ground_truth": "44",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 33,
"problem": "A non-self intersecting hexagon $R A N D O M$ is formed by assigning the labels $R, A, N, D, O, M$ in some order to the points\n$$\n(0,0),(10,0),(10,10),(0,10),(3,4),(6,2) .\n$$\n\nLet $a_{\\text {max }}$ be the greatest possible area of $R A N D O M$ and $a_{\\text {min }}$ the least pos... |
[
{
"content": "Suppose we have a uniformly random function from $\\{1,2,3, \\ldots, 25\\}$ to itself. Find the expected value of\n\n$$\n\\sum_{x=1}^{25}(f(f(x))-x)^{2}\n$$",
"role": "user"
}
] | {
"ground_truth": "2496",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 34,
"problem": "Suppose we have a uniformly random function from $\\{1,2,3, \\ldots, 25\\}$ to itself. Find the expected value of\n\n$$\n\\sum_{x=1}^{25}(f(f(x))-x)^{2}\n$$"
} |
[
{
"content": "Suppose we have a regular $24$-gon labeled $A_{1} \\cdots A_{24}$. We will draw $2$ smaller regular $24$-gons within $A_{1} \\cdots A_{24}$. For the sake of this problem, make $A_{i}=A_{i+24}$.\nWith our first configuration, we create $3$ stars by creating lines $\\overline{A_{i} A_{i+9}}$. A $24$... | {
"ground_truth": "3+\\sqrt{2}+\\sqrt{3}+\\sqrt{6}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 35,
"problem": "Suppose we have a regular $24$-gon labeled $A_{1} \\cdots A_{24}$. We will draw $2$ smaller regular $24$-gons within $A_{1} \\cdots A_{24}$. For the sake of this problem, make $A_{i}=A_{i+24}$.\nWith our first configuration, we create $3$ stars by creating lines $\\overline{A_{i} A_{i+9}}... |
[
{
"content": "The binomial coefficient $\\binom{n}{k}$ can be defined as the coefficient of $x^{k}$ in the expansion of $(1+x)^{n}$. Similarly, define the trinomial coefficient $\\binom{n}{k}_{3}$ as the coefficient of $x^{k}$ in the expansion of $(1+x+$ $\\left.x^{2}\\right)^{n}$.\nDetermine the number of inte... | {
"ground_truth": "648",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 36,
"problem": "The binomial coefficient $\\binom{n}{k}$ can be defined as the coefficient of $x^{k}$ in the expansion of $(1+x)^{n}$. Similarly, define the trinomial coefficient $\\binom{n}{k}_{3}$ as the coefficient of $x^{k}$ in the expansion of $(1+x+$ $\\left.x^{2}\\right)^{n}$.\nDetermine the numbe... |
[
{
"content": "Let $U$ be the set of all possible complex numbers $m$ so that the $4$ roots of $\\left(x^{2}+2 x+5\\right)\\left(x^{2}-\\right.$ $2 m x+25)=0$ are concyclic in the complex plane. One can show that when the points of $U$ are plotted on the complex plane, it is visualized as the finite union of som... | {
"ground_truth": "10+\\frac{40\\pi}{3}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 37,
"problem": "Let $U$ be the set of all possible complex numbers $m$ so that the $4$ roots of $\\left(x^{2}+2 x+5\\right)\\left(x^{2}-\\right.$ $2 m x+25)=0$ are concyclic in the complex plane. One can show that when the points of $U$ are plotted on the complex plane, it is visualized as the finite uni... |
[
{
"content": "Given a triangle, $A B=78, B C=50, A C=112$, construct squares $A B X Y, B C P Q, A C M N$ outside the triangle. Let $L_{1}, L_{2}, L_{3}$ be the midpoints of $\\overline{M P}, \\overline{Q X}, \\overline{N Y}$, respectively. Find the area of $L_{1} L_{2} L_{3}$.",
"role": "user"
}
] | {
"ground_truth": "8222",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 38,
"problem": "Given a triangle, $A B=78, B C=50, A C=112$, construct squares $A B X Y, B C P Q, A C M N$ outside the triangle. Let $L_{1}, L_{2}, L_{3}$ be the midpoints of $\\overline{M P}, \\overline{Q X}, \\overline{N Y}$, respectively. Find the area of $L_{1} L_{2} L_{3}$."
} |
[
{
"content": "In a $2024 \\times 2024$ grid of squares, each square is colored either black or white. An ant starts at some black square in the grid and starts walking parallel to the sides of the grid. During this walk, it can choose (not required) to turn $90^{\\circ}$ clockwise or counterclockwise if it is c... | {
"ground_truth": "2024^{4046}",
"style": "rule-lighteval/MATH_v2"
} | cmimc25 | MATH | {
"index": 39,
"problem": "In a $2024 \\times 2024$ grid of squares, each square is colored either black or white. An ant starts at some black square in the grid and starts walking parallel to the sides of the grid. During this walk, it can choose (not required) to turn $90^{\\circ}$ clockwise or counterclockwise i... |
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