data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
olympiad_bench_boxed | [
{
"content": "Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding 5000. Then she fixes 20 distinct positive integers $a_{1}, a_{2}, \\ldots, a_{20}$ such that, for each $k=1,2, \\ldots, 20$, the numbers $N$ and $a_{k}$ are congruent modulo $k$. By a move, Sergey tells ... | MATH | {
"ground_truth": [
"2"
],
"style": "rule"
} | {
"index": 0,
"raw_problem": "Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding 5000. Then she fixes 20 distinct positive integers $a_{1}, a_{2}, \\ldots, a_{20}$ such that, for each $k=1,2, \\ldots, 20$, the numbers $N$ and $a_{k}$ are congruent modulo $k$. By a move, S... |
olympiad_bench_boxed | [
{
"content": "Given a positive integer $n$, determine the largest real number $\\mu$ satisfying the following condition: for every $4 n$-point configuration $C$ in an open unit square $U$, there exists an open rectangle in $U$, whose sides are parallel to those of $U$, which contains exactly one point of $C$, a... | MATH | {
"ground_truth": [
"$\\frac{1}{2 n+2}$"
],
"style": "rule"
} | {
"index": 1,
"raw_problem": "Given a positive integer $n$, determine the largest real number $\\mu$ satisfying the following condition: for every $4 n$-point configuration $C$ in an open unit square $U$, there exists an open rectangle in $U$, whose sides are parallel to those of $U$, which contains exactly one poi... |
olympiad_bench_boxed | [
{
"content": "\nFind (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$2^{1009}$"
],
"style": "rule"
} | {
"index": 2,
"raw_problem": "\nFind (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k \\leq n$, and $k+1$ distinct integers $x_{1}, x_{2}, \\ldots, x_{k+1}$ such that\n\n\n\n$$\n\nP\\left(x_{1}\\ri... | MATH | {
"ground_truth": [
"2"
],
"style": "rule"
} | {
"index": 3,
"raw_problem": "Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k \\leq n$, and $k+1$ distinct integers $x_{1}, x_{2}, \\ldots, x_{k+1}$ such that\n\n\n\n$$\n\nP\\le... |
olympiad_bench_boxed | [
{
"content": "Let $n$ be an integer greater than 1 and let $X$ be an $n$-element set. A non-empty collection of subsets $A_{1}, \\ldots, A_{k}$ of $X$ is tight if the union $A_{1} \\cup \\cdots \\cup A_{k}$ is a proper subset of $X$ and no element of $X$ lies in exactly one of the $A_{i}$ s. Find the largest ca... | MATH | {
"ground_truth": [
"$2n-2$"
],
"style": "rule"
} | {
"index": 4,
"raw_problem": "Let $n$ be an integer greater than 1 and let $X$ be an $n$-element set. A non-empty collection of subsets $A_{1}, \\ldots, A_{k}$ of $X$ is tight if the union $A_{1} \\cup \\cdots \\cup A_{k}$ is a proper subset of $X$ and no element of $X$ lies in exactly one of the $A_{i}$ s. Find th... |
olympiad_bench_boxed | [
{
"content": "Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $x^{3}+y^{3}=$ $p(x y+p)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$(1,8,19), (2,7,13), (4,5,7)$"
],
"style": "rule"
} | {
"index": 5,
"raw_problem": "Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $x^{3}+y^{3}=$ $p(x y+p)$.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Let $n \\geqslant 2$ be an integer, and let $f$ be a $4 n$-variable polynomial with real coefficients. Assume that, for any $2 n$ points $\\left(x_{1}, y_{1}\\right), \\ldots,\\left(x_{2 n}, y_{2 n}\\right)$ in the plane, $f\\left(x_{1}, y_{1}, \\ldots, x_{2 n}, y_{2 n}\\right)=0$ if and only if t... | MATH | {
"ground_truth": [
"$2n$"
],
"style": "rule"
} | {
"index": 6,
"raw_problem": "Let $n \\geqslant 2$ be an integer, and let $f$ be a $4 n$-variable polynomial with real coefficients. Assume that, for any $2 n$ points $\\left(x_{1}, y_{1}\\right), \\ldots,\\left(x_{2 n}, y_{2 n}\\right)$ in the plane, $f\\left(x_{1}, y_{1}, \\ldots, x_{2 n}, y_{2 n}\\right)=0$ if a... |
olympiad_bench_boxed | [
{
"content": "For a positive integer $a$, define a sequence of integers $x_{1}, x_{2}, \\ldots$ by letting $x_{1}=a$ and $x_{n+1}=2 x_{n}+1$ for $n \\geq 1$. Let $y_{n}=2^{x_{n}}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_{1}, \\ldots, y_{k}$ are all prime. L... | MATH | {
"ground_truth": [
"2"
],
"style": "rule"
} | {
"index": 7,
"raw_problem": "For a positive integer $a$, define a sequence of integers $x_{1}, x_{2}, \\ldots$ by letting $x_{1}=a$ and $x_{n+1}=2 x_{n}+1$ for $n \\geq 1$. Let $y_{n}=2^{x_{n}}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_{1}, \\ldots, y_{k}$ are ... |
olympiad_bench_boxed | [
{
"content": "Let $n$ be a positive integer and fix $2 n$ distinct points on a circumference. Split these points into $n$ pairs and join the points in each pair by an arrow (i.e., an oriented line segment). The resulting configuration is good if no two arrows cross, and there are no arrows $\\overrightarrow{A B... | MATH | {
"ground_truth": [
"$\\binom{2n}{n}$"
],
"style": "rule"
} | {
"index": 8,
"raw_problem": "Let $n$ be a positive integer and fix $2 n$ distinct points on a circumference. Split these points into $n$ pairs and join the points in each pair by an arrow (i.e., an oriented line segment). The resulting configuration is good if no two arrows cross, and there are no arrows $\\overri... |
olympiad_bench_boxed | [
{
"content": "Given positive integers $m$ and $n \\geq m$, determine the largest number of dominoes $(1 \\times 2$ or $2 \\times 1$ rectangles) that can be placed on a rectangular board with $m$ rows and $2 n$ columns consisting of cells $(1 \\times 1$ squares $)$ so that:\n\n\n\n(i) each domino covers exactly ... | MATH | {
"ground_truth": [
"$m n-\\lfloor m / 2\\rfloor$"
],
"style": "rule"
} | {
"index": 9,
"raw_problem": "Given positive integers $m$ and $n \\geq m$, determine the largest number of dominoes $(1 \\times 2$ or $2 \\times 1$ rectangles) that can be placed on a rectangular board with $m$ rows and $2 n$ columns consisting of cells $(1 \\times 1$ squares $)$ so that:\n\n\n\n(i) each domino cov... |
olympiad_bench_boxed | [
{
"content": "A cubic sequence is a sequence of integers given by $a_{n}=n^{3}+b n^{2}+c n+d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers.\n\nDetermine the possible values of $a_{2015} \\cdot a_{2016}$ for a cubic sequence satisfying the condition in... | MATH | {
"ground_truth": [
"0"
],
"style": "rule"
} | {
"index": 10,
"raw_problem": "A cubic sequence is a sequence of integers given by $a_{n}=n^{3}+b n^{2}+c n+d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers.\n\nDetermine the possible values of $a_{2015} \\cdot a_{2016}$ for a cubic sequence satisfying the... |
olympiad_bench_boxed | [
{
"content": "Find all functions $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$ such that\n\n$$\nf(x+f(y))=f(x+y)+f(y)\\tag{1}\n$$\n\nfor all $x, y \\in \\mathbb{R}^{+}$. (Symbol $\\mathbb{R}^{+}$denotes the set of all positive real numbers.) Let's think step by step and output the final answer within \\boxe... | MATH | {
"ground_truth": [
"$f(x)=2 x$"
],
"style": "rule"
} | {
"index": 11,
"raw_problem": "Find all functions $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$ such that\n\n$$\nf(x+f(y))=f(x+y)+f(y)\\tag{1}\n$$\n\nfor all $x, y \\in \\mathbb{R}^{+}$. (Symbol $\\mathbb{R}^{+}$denotes the set of all positive real numbers.)",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Let $n>1$ be an integer. In the space, consider the set\n$$\nS=\\{(x, y, z) \\mid x, y, z \\in\\{0,1, \\ldots, n\\}, x+y+z>0\\}\n$$\nFind the smallest number of planes that jointly contain all $(n+1)^{3}-1$ points of $S$ but none of them passes through the origin. Let's think step by step and outp... | MATH | {
"ground_truth": [
"$3 n$"
],
"style": "rule"
} | {
"index": 12,
"raw_problem": "Let $n>1$ be an integer. In the space, consider the set\n$$\nS=\\{(x, y, z) \\mid x, y, z \\in\\{0,1, \\ldots, n\\}, x+y+z>0\\}\n$$\nFind the smallest number of planes that jointly contain all $(n+1)^{3}-1$ points of $S$ but none of them passes through the origin.",
"split": "test"
... |
olympiad_bench_boxed | [
{
"content": "Find all positive integers $n$, for which the numbers in the set $S=\\{1,2, \\ldots, n\\}$ can be colored red and blue, with the following condition being satisfied: the set $S \\times S \\times S$ contains exactly 2007 ordered triples $(x, y, z)$ such that (i) $x, y, z$ are of the same color and ... | MATH | {
"ground_truth": [
"$69$,$84$"
],
"style": "rule"
} | {
"index": 13,
"raw_problem": "Find all positive integers $n$, for which the numbers in the set $S=\\{1,2, \\ldots, n\\}$ can be colored red and blue, with the following condition being satisfied: the set $S \\times S \\times S$ contains exactly 2007 ordered triples $(x, y, z)$ such that (i) $x, y, z$ are of the sa... |
olympiad_bench_boxed | [
{
"content": "Determine the smallest positive real number $k$ with the following property.\n\nLet $A B C D$ be a convex quadrilateral, and let points $A_{1}, B_{1}, C_{1}$ and $D_{1}$ lie on sides $A B, B C$, $C D$ and $D A$, respectively. Consider the areas of triangles $A A_{1} D_{1}, B B_{1} A_{1}, C C_{1} B... | MATH | {
"ground_truth": [
"$k=1$"
],
"style": "rule"
} | {
"index": 14,
"raw_problem": "Determine the smallest positive real number $k$ with the following property.\n\nLet $A B C D$ be a convex quadrilateral, and let points $A_{1}, B_{1}, C_{1}$ and $D_{1}$ lie on sides $A B, B C$, $C D$ and $D A$, respectively. Consider the areas of triangles $A A_{1} D_{1}, B B_{1} A_{... |
olympiad_bench_boxed | [
{
"content": "Find all pairs $(k, n)$ of positive integers for which $7^{k}-3^{n}$ divides $k^{4}+n^{2}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$(2,4)$"
],
"style": "rule"
} | {
"index": 15,
"raw_problem": "Find all pairs $(k, n)$ of positive integers for which $7^{k}-3^{n}$ divides $k^{4}+n^{2}$.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Find all surjective functions $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ such that for every $m, n \\in \\mathbb{N}$ and every prime $p$, the number $f(m+n)$ is divisible by $p$ if and only if $f(m)+f(n)$ is divisible by $p$.\n\n( $\\mathbb{N}$ is the set of all positive integers.) Let's think step... | MATH | {
"ground_truth": [
"$f(n)=n$"
],
"style": "rule"
} | {
"index": 16,
"raw_problem": "Find all surjective functions $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ such that for every $m, n \\in \\mathbb{N}$ and every prime $p$, the number $f(m+n)$ is divisible by $p$ if and only if $f(m)+f(n)$ is divisible by $p$.\n\n( $\\mathbb{N}$ is the set of all positive integers.)",
... |
olympiad_bench_boxed | [
{
"content": "Determine all pairs $(f, g)$ of functions from the set of positive integers to itself that satisfy\n\n$$\nf^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1\n$$\n\nfor every positive integer $n$. Here, $f^{k}(n)$ means $\\underbrace{f(f(\\ldots f}_{k}(n) \\ldots))$. Let's think step by step and output the ... | MATH | {
"ground_truth": [
"$f(n)=n$, $g(n)=1$"
],
"style": "rule"
} | {
"index": 17,
"raw_problem": "Determine all pairs $(f, g)$ of functions from the set of positive integers to itself that satisfy\n\n$$\nf^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1\n$$\n\nfor every positive integer $n$. Here, $f^{k}(n)$ means $\\underbrace{f(f(\\ldots f}_{k}(n) \\ldots))$.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_{1}, A_{2}, \\ldots, A_{k}$ such that for all integers $n \\geq 15$ and all $i \\in\\{1,2, \\ldots, k\\}$ there exist two distinct elements of $A_{... | MATH | {
"ground_truth": [
"3"
],
"style": "rule"
} | {
"index": 18,
"raw_problem": "Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_{1}, A_{2}, \\ldots, A_{k}$ such that for all integers $n \\geq 15$ and all $i \\in\\{1,2, \\ldots, k\\}$ there exist two distinct ele... |
olympiad_bench_boxed | [
{
"content": "Let $m$ be a positive integer and consider a checkerboard consisting of $m$ by $m$ unit squares. At the midpoints of some of these unit squares there is an ant. At time 0, each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite directions mee... | MATH | {
"ground_truth": [
"$\\frac{3 m}{2}-1$"
],
"style": "rule"
} | {
"index": 19,
"raw_problem": "Let $m$ be a positive integer and consider a checkerboard consisting of $m$ by $m$ unit squares. At the midpoints of some of these unit squares there is an ant. At time 0, each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite d... |
olympiad_bench_boxed | [
{
"content": "On a square table of 2011 by 2011 cells we place a finite number of napkins that each cover a square of 52 by 52 cells. In each cell we write the number of napkins covering it, and we record the maximal number $k$ of cells that all contain the same nonzero number. Considering all possible napkin c... | MATH | {
"ground_truth": [
"3986729"
],
"style": "rule"
} | {
"index": 20,
"raw_problem": "On a square table of 2011 by 2011 cells we place a finite number of napkins that each cover a square of 52 by 52 cells. In each cell we write the number of napkins covering it, and we record the maximal number $k$ of cells that all contain the same nonzero number. Considering all poss... |
olympiad_bench_boxed | [
{
"content": "For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k$. Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the differences\n\n$$\nt(n+a)-t(n), \\quad t(n+a+1)-t(n+1), \\quad \\ldots, \\quad t(n+2 a-1)-t(n+a-1)\n$$\n\nare divisible b... | MATH | {
"ground_truth": [
"1,3,5"
],
"style": "rule"
} | {
"index": 21,
"raw_problem": "For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k$. Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the differences\n\n$$\nt(n+a)-t(n), \\quad t(n+a+1)-t(n+1), \\quad \\ldots, \\quad t(n+2 a-1)-t(n+a-1)\n$$\n\nar... |
olympiad_bench_boxed | [
{
"content": "Let $x_{1}, \\ldots, x_{100}$ be nonnegative real numbers such that $x_{i}+x_{i+1}+x_{i+2} \\leq 1$ for all $i=1, \\ldots, 100$ (we put $x_{101}=x_{1}, x_{102}=x_{2}$ ). Find the maximal possible value of the sum\n\n$$\nS=\\sum_{i=1}^{100} x_{i} x_{i+2}\n$$ Let's think step by step and output the ... | MATH | {
"ground_truth": [
"$\\frac{25}{2}$"
],
"style": "rule"
} | {
"index": 22,
"raw_problem": "Let $x_{1}, \\ldots, x_{100}$ be nonnegative real numbers such that $x_{i}+x_{i+1}+x_{i+2} \\leq 1$ for all $i=1, \\ldots, 100$ (we put $x_{101}=x_{1}, x_{102}=x_{2}$ ). Find the maximal possible value of the sum\n\n$$\nS=\\sum_{i=1}^{100} x_{i} x_{i+2}\n$$",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Denote by $\\mathbb{Q}^{+}$the set of all positive rational numbers. Determine all functions $f: \\mathbb{Q}^{+} \\rightarrow \\mathbb{Q}^{+}$ which satisfy the following equation for all $x, y \\in \\mathbb{Q}^{+}$:\n\n$$\nf\\left(f(x)^{2} y\\right)=x^{3} f(x y)\n\\tag{1}\n$$ Let's think step by ... | MATH | {
"ground_truth": [
"$f(x)=\\frac{1}{x}$"
],
"style": "rule"
} | {
"index": 23,
"raw_problem": "Denote by $\\mathbb{Q}^{+}$the set of all positive rational numbers. Determine all functions $f: \\mathbb{Q}^{+} \\rightarrow \\mathbb{Q}^{+}$ which satisfy the following equation for all $x, y \\in \\mathbb{Q}^{+}$:\n\n$$\nf\\left(f(x)^{2} y\\right)=x^{3} f(x y)\n\\tag{1}\n$$",
"sp... |
olympiad_bench_boxed | [
{
"content": "On some planet, there are $2^{N}$ countries $(N \\geq 4)$. Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \\times 1$, each field being either yellow or blue. No two countries have the same flag.\n\nWe say that a set of $N$ flags is diverse if these flags... | MATH | {
"ground_truth": [
"$M=2^{N-2}+1$"
],
"style": "rule"
} | {
"index": 24,
"raw_problem": "On some planet, there are $2^{N}$ countries $(N \\geq 4)$. Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \\times 1$, each field being either yellow or blue. No two countries have the same flag.\n\nWe say that a set of $N$ flags is diverse i... |
olympiad_bench_boxed | [
{
"content": "2500 chess kings have to be placed on a $100 \\times 100$ chessboard so that\n\n(i) no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);\n\n(ii) each row and each column contains exactly 25 kings.\n\nFind the number of such arrangements. (Two arr... | MATH | {
"ground_truth": [
"2"
],
"style": "rule"
} | {
"index": 25,
"raw_problem": "2500 chess kings have to be placed on a $100 \\times 100$ chessboard so that\n\n(i) no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);\n\n(ii) each row and each column contains exactly 25 kings.\n\nFind the number of such arrangeme... |
olympiad_bench_boxed | [
{
"content": "Find the least positive integer $n$ for which there exists a set $\\left\\{s_{1}, s_{2}, \\ldots, s_{n}\\right\\}$ consisting of $n$ distinct positive integers such that\n\n$$\n\\left(1-\\frac{1}{s_{1}}\\right)\\left(1-\\frac{1}{s_{2}}\\right) \\ldots\\left(1-\\frac{1}{s_{n}}\\right)=\\frac{51}{20... | MATH | {
"ground_truth": [
"39"
],
"style": "rule"
} | {
"index": 26,
"raw_problem": "Find the least positive integer $n$ for which there exists a set $\\left\\{s_{1}, s_{2}, \\ldots, s_{n}\\right\\}$ consisting of $n$ distinct positive integers such that\n\n$$\n\\left(1-\\frac{1}{s_{1}}\\right)\\left(1-\\frac{1}{s_{2}}\\right) \\ldots\\left(1-\\frac{1}{s_{n}}\\right)=... |
olympiad_bench_boxed | [
{
"content": "Find all pairs $(m, n)$ of nonnegative integers for which\n\n$$\nm^{2}+2 \\cdot 3^{n}=m\\left(2^{n+1}-1\\right)\n\\tag{1}\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$(6,3),(9,3),(9,5),(54,5)$"
],
"style": "rule"
} | {
"index": 27,
"raw_problem": "Find all pairs $(m, n)$ of nonnegative integers for which\n\n$$\nm^{2}+2 \\cdot 3^{n}=m\\left(2^{n+1}-1\\right)\n\\tag{1}\n$$",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Find the smallest number $n$ such that there exist polynomials $f_{1}, f_{2}, \\ldots, f_{n}$ with rational coefficients satisfying\n\n$$\nx^{2}+7=f_{1}(x)^{2}+f_{2}(x)^{2}+\\cdots+f_{n}(x)^{2} .\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"5"
],
"style": "rule"
} | {
"index": 28,
"raw_problem": "Find the smallest number $n$ such that there exist polynomials $f_{1}, f_{2}, \\ldots, f_{n}$ with rational coefficients satisfying\n\n$$\nx^{2}+7=f_{1}(x)^{2}+f_{2}(x)^{2}+\\cdots+f_{n}(x)^{2} .\n$$",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Determine the smallest number $M$ such that the inequality\n\n$$\n\\left|a b\\left(a^{2}-b^{2}\\right)+b c\\left(b^{2}-c^{2}\\right)+c a\\left(c^{2}-a^{2}\\right)\\right| \\leq M\\left(a^{2}+b^{2}+c^{2}\\right)^{2}\n$$\n\nholds for all real numbers $a, b, c$. Let's think step by step and output th... | MATH | {
"ground_truth": [
"$M=\\frac{9}{32} \\sqrt{2}$"
],
"style": "rule"
} | {
"index": 29,
"raw_problem": "Determine the smallest number $M$ such that the inequality\n\n$$\n\\left|a b\\left(a^{2}-b^{2}\\right)+b c\\left(b^{2}-c^{2}\\right)+c a\\left(c^{2}-a^{2}\\right)\\right| \\leq M\\left(a^{2}+b^{2}+c^{2}\\right)^{2}\n$$\n\nholds for all real numbers $a, b, c$.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals.\n\nSuppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possi... | MATH | {
"ground_truth": [
"$1003$"
],
"style": "rule"
} | {
"index": 30,
"raw_problem": "A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals.\n\nSuppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the ... |
olympiad_bench_boxed | [
{
"content": "In triangle $A B C$, let $J$ be the centre of the excircle tangent to side $B C$ at $A_{1}$ and to the extensions of sides $A C$ and $A B$ at $B_{1}$ and $C_{1}$, respectively. Suppose that the lines $A_{1} B_{1}$ and $A B$ are perpendicular and intersect at $D$. Let $E$ be the foot of the perpend... | MATH | {
"ground_truth": [
"$\\angle B E A_{1}=90$,$\\angle A E B_{1}=90$"
],
"style": "rule"
} | {
"index": 31,
"raw_problem": "In triangle $A B C$, let $J$ be the centre of the excircle tangent to side $B C$ at $A_{1}$ and to the extensions of sides $A C$ and $A B$ at $B_{1}$ and $C_{1}$, respectively. Suppose that the lines $A_{1} B_{1}$ and $A B$ are perpendicular and intersect at $D$. Let $E$ be the foot o... |
olympiad_bench_boxed | [
{
"content": "Determine all pairs $(x, y)$ of integers satisfying the equation\n\n$$\n1+2^{x}+2^{2 x+1}=y^{2}\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$(0,2),(0,-2),(4,23),(4,-23)$"
],
"style": "rule"
} | {
"index": 32,
"raw_problem": "Determine all pairs $(x, y)$ of integers satisfying the equation\n\n$$\n1+2^{x}+2^{2 x+1}=y^{2}\n$$",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Given a positive integer $n$, find the smallest value of $\\left\\lfloor\\frac{a_{1}}{1}\\right\\rfloor+\\left\\lfloor\\frac{a_{2}}{2}\\right\\rfloor+\\cdots+\\left\\lfloor\\frac{a_{n}}{n}\\right\\rfloor$ over all permutations $\\left(a_{1}, a_{2}, \\ldots, a_{n}\\right)$ of $(1,2, \\ldots, n)$. L... | MATH | {
"ground_truth": [
"$\\left\\lfloor\\log _{2} n\\right\\rfloor+1$"
],
"style": "rule"
} | {
"index": 33,
"raw_problem": "Given a positive integer $n$, find the smallest value of $\\left\\lfloor\\frac{a_{1}}{1}\\right\\rfloor+\\left\\lfloor\\frac{a_{2}}{2}\\right\\rfloor+\\cdots+\\left\\lfloor\\frac{a_{n}}{n}\\right\\rfloor$ over all permutations $\\left(a_{1}, a_{2}, \\ldots, a_{n}\\right)$ of $(1,2, \\... |
olympiad_bench_boxed | [
{
"content": "Let $n \\geqslant 3$ be an integer. An integer $m \\geqslant n+1$ is called $n$-colourful if, given infinitely many marbles in each of $n$ colours $C_{1}, C_{2}, \\ldots, C_{n}$, it is possible to place $m$ of them around a circle so that in any group of $n+1$ consecutive marbles there is at least... | MATH | {
"ground_truth": [
"$m_{\\max }=n^{2}-n-1$"
],
"style": "rule"
} | {
"index": 34,
"raw_problem": "Let $n \\geqslant 3$ be an integer. An integer $m \\geqslant n+1$ is called $n$-colourful if, given infinitely many marbles in each of $n$ colours $C_{1}, C_{2}, \\ldots, C_{n}$, it is possible to place $m$ of them around a circle so that in any group of $n+1$ consecutive marbles ther... |
olympiad_bench_boxed | [
{
"content": "Determine the largest $N$ for which there exists a table $T$ of integers with $N$ rows and 100 columns that has the following properties:\n\n(i) Every row contains the numbers 1,2, ., 100 in some order.\n\n(ii) For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r, c)-T(s, c... | MATH | {
"ground_truth": [
"$\\frac{100!}{2^{50}}$"
],
"style": "rule"
} | {
"index": 35,
"raw_problem": "Determine the largest $N$ for which there exists a table $T$ of integers with $N$ rows and 100 columns that has the following properties:\n\n(i) Every row contains the numbers 1,2, ., 100 in some order.\n\n(ii) For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T... |
olympiad_bench_boxed | [
{
"content": "Determine all integers $n \\geqslant 1$ for which there exists a pair of positive integers $(a, b)$ such that no cube of a prime divides $a^{2}+b+3$ and\n\n$$\n\\frac{a b+3 b+8}{a^{2}+b+3}=n\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"2"
],
"style": "rule"
} | {
"index": 36,
"raw_problem": "Determine all integers $n \\geqslant 1$ for which there exists a pair of positive integers $(a, b)$ such that no cube of a prime divides $a^{2}+b+3$ and\n\n$$\n\\frac{a b+3 b+8}{a^{2}+b+3}=n\n$$",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $\\left(d_{1}, d_{2}, \\ldots, d_{k}\\right)$ such that for every $i=1,2, \\ldots, k$, the number $d_{1}+\\cdots+d_{i}$ is a perfect square. Let's think step by step and output the final... | MATH | {
"ground_truth": [
"1,3"
],
"style": "rule"
} | {
"index": 37,
"raw_problem": "Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $\\left(d_{1}, d_{2}, \\ldots, d_{k}\\right)$ such that for every $i=1,2, \\ldots, k$, the number $d_{1}+\\cdots+d_{i}$ is a perfect square.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:\n\n- In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers... | MATH | {
"ground_truth": [
"$-2,0,2$"
],
"style": "rule"
} | {
"index": 38,
"raw_problem": "Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:\n\n- In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily dist... |
olympiad_bench_boxed | [
{
"content": "An integer $n \\geqslant 3$ is given. We call an $n$-tuple of real numbers $\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right)$ Shiny if for each permutation $y_{1}, y_{2}, \\ldots, y_{n}$ of these numbers we have\n\n$$\n\\sum_{i=1}^{n-1} y_{i} y_{i+1}=y_{1} y_{2}+y_{2} y_{3}+y_{3} y_{4}+\\cdots+y_{n-1} ... | MATH | {
"ground_truth": [
"$-(n-1) / 2$"
],
"style": "rule"
} | {
"index": 39,
"raw_problem": "An integer $n \\geqslant 3$ is given. We call an $n$-tuple of real numbers $\\left(x_{1}, x_{2}, \\ldots, x_{n}\\right)$ Shiny if for each permutation $y_{1}, y_{2}, \\ldots, y_{n}$ of these numbers we have\n\n$$\n\\sum_{i=1}^{n-1} y_{i} y_{i+1}=y_{1} y_{2}+y_{2} y_{3}+y_{3} y_{4}+\\c... |
olympiad_bench_boxed | [
{
"content": "Let $n>1$ be an integer. An $n \\times n \\times n$ cube is composed of $n^{3}$ unit cubes. Each unit cube is painted with one color. For each $n \\times n \\times 1$ box consisting of $n^{2}$ unit cubes (of any of the three possible orientations), we consider the set of the colors present in that... | MATH | {
"ground_truth": [
"$\\frac{n(n+1)(2 n+1)}{6}$"
],
"style": "rule"
} | {
"index": 40,
"raw_problem": "Let $n>1$ be an integer. An $n \\times n \\times n$ cube is composed of $n^{3}$ unit cubes. Each unit cube is painted with one color. For each $n \\times n \\times 1$ box consisting of $n^{2}$ unit cubes (of any of the three possible orientations), we consider the set of the colors pr... |
olympiad_bench_boxed | [
{
"content": "Let $n$ be a given positive integer. In the Cartesian plane, each lattice point with nonnegative coordinates initially contains a butterfly, and there are no other butterflies. The neighborhood of a lattice point $c$ consists of all lattice points within the axis-aligned $(2 n+1) \\times$ $(2 n+1)... | MATH | {
"ground_truth": [
"$n^{2}+1$"
],
"style": "rule"
} | {
"index": 41,
"raw_problem": "Let $n$ be a given positive integer. In the Cartesian plane, each lattice point with nonnegative coordinates initially contains a butterfly, and there are no other butterflies. The neighborhood of a lattice point $c$ consists of all lattice points within the axis-aligned $(2 n+1) \\ti... |
olympiad_bench_boxed | [
{
"content": "There are 2017 mutually external circles drawn on a blackboard, such that no two are tangent and no three share a common tangent. A tangent segment is a line segment that is a common tangent to two circles, starting at one tangent point and ending at the other one. Luciano is drawing tangent segme... | MATH | {
"ground_truth": [
"6048"
],
"style": "rule"
} | {
"index": 42,
"raw_problem": "There are 2017 mutually external circles drawn on a blackboard, such that no two are tangent and no three share a common tangent. A tangent segment is a line segment that is a common tangent to two circles, starting at one tangent point and ending at the other one. Luciano is drawing ... |
olympiad_bench_boxed | [
{
"content": "Call a rational number short if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m$-tastic if there exists a number $c \\in\\{1,2,3, \\ldots, 2017\\}$ such that $\\frac{10^{t}-1}{c \\cdot m}$ is short, and such that $\\frac{10... | MATH | {
"ground_truth": [
"807"
],
"style": "rule"
} | {
"index": 43,
"raw_problem": "Call a rational number short if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m$-tastic if there exists a number $c \\in\\{1,2,3, \\ldots, 2017\\}$ such that $\\frac{10^{t}-1}{c \\cdot m}$ is short, and such th... |
olympiad_bench_boxed | [
{
"content": "Find all pairs $(p, q)$ of prime numbers with $p>q$ for which the number\n\n$$\n\\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}\n$$\n\nis an integer. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$(3,2)$"
],
"style": "rule"
} | {
"index": 44,
"raw_problem": "Find all pairs $(p, q)$ of prime numbers with $p>q$ for which the number\n\n$$\n\\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}\n$$\n\nis an integer.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Find the smallest positive integer $n$, or show that no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $\\left(a_{1}, a_{2}, \\ldots, a_{n}\\right)$ such that both\n\n$$\na_{1}+a_{2}+\\cdots+a_{n} \\quad \\text { and } \\qua... | MATH | {
"ground_truth": [
"3"
],
"style": "rule"
} | {
"index": 45,
"raw_problem": "Find the smallest positive integer $n$, or show that no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $\\left(a_{1}, a_{2}, \\ldots, a_{n}\\right)$ such that both\n\n$$\na_{1}+a_{2}+\\cdots+a_{n} \\quad \\text ... |
olympiad_bench_boxed | [
{
"content": "Find the smallest real constant $C$ such that for any positive real numbers $a_{1}, a_{2}, a_{3}, a_{4}$ and $a_{5}$ (not necessarily distinct), one can always choose distinct subscripts $i, j, k$ and $l$ such that\n\n$$\n\\left|\\frac{a_{i}}{a_{j}}-\\frac{a_{k}}{a_{l}}\\right| \\leqslant C .\\tag... | MATH | {
"ground_truth": [
"$\\frac{1}{2}$"
],
"style": "rule"
} | {
"index": 46,
"raw_problem": "Find the smallest real constant $C$ such that for any positive real numbers $a_{1}, a_{2}, a_{3}, a_{4}$ and $a_{5}$ (not necessarily distinct), one can always choose distinct subscripts $i, j, k$ and $l$ such that\n\n$$\n\\left|\\frac{a_{i}}{a_{j}}-\\frac{a_{k}}{a_{l}}\\right| \\leqs... |
olympiad_bench_boxed | [
{
"content": "The equation\n\n$$\n(x-1)(x-2) \\cdots(x-2016)=(x-1)(x-2) \\cdots(x-2016)\n$$\n\nis written on the board. One tries to erase some linear factors from both sides so that each side still has at least one factor, and the resulting equation has no real roots. Find the least number of linear factors on... | MATH | {
"ground_truth": [
"2016"
],
"style": "rule"
} | {
"index": 47,
"raw_problem": "The equation\n\n$$\n(x-1)(x-2) \\cdots(x-2016)=(x-1)(x-2) \\cdots(x-2016)\n$$\n\nis written on the board. One tries to erase some linear factors from both sides so that each side still has at least one factor, and the resulting equation has no real roots. Find the least number of line... |
olympiad_bench_boxed | [
{
"content": "Determine the largest real number $a$ such that for all $n \\geqslant 1$ and for all real numbers $x_{0}, x_{1}, \\ldots, x_{n}$ satisfying $0=x_{0}<x_{1}<x_{2}<\\cdots<x_{n}$, we have\n\n$$\n\\frac{1}{x_{1}-x_{0}}+\\frac{1}{x_{2}-x_{1}}+\\cdots+\\frac{1}{x_{n}-x_{n-1}} \\geqslant a\\left(\\frac{2... | MATH | {
"ground_truth": [
"$\\frac{4}{9}$"
],
"style": "rule"
} | {
"index": 48,
"raw_problem": "Determine the largest real number $a$ such that for all $n \\geqslant 1$ and for all real numbers $x_{0}, x_{1}, \\ldots, x_{n}$ satisfying $0=x_{0}<x_{1}<x_{2}<\\cdots<x_{n}$, we have\n\n$$\n\\frac{1}{x_{1}-x_{0}}+\\frac{1}{x_{2}-x_{1}}+\\cdots+\\frac{1}{x_{n}-x_{n-1}} \\geqslant a\\... |
olympiad_bench_boxed | [
{
"content": "Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints:\n\n- each cell contains a distinct divisor;\n- the sums of all rows are equal; and\n- the sums of all columns are equal. Let's think step by step ... | MATH | {
"ground_truth": [
"1"
],
"style": "rule"
} | {
"index": 49,
"raw_problem": "Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints:\n\n- each cell contains a distinct divisor;\n- the sums of all rows are equal; and\n- the sums of all columns are equal.",
"split":... |
olympiad_bench_boxed | [
{
"content": "Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2 n \\times 2 n$ board so that there exists a unique partition of the board into $1 \\times 2$ and $2 \\times 1$ dominoes, none of which contains two marke... | MATH | {
"ground_truth": [
"$2 n$"
],
"style": "rule"
} | {
"index": 50,
"raw_problem": "Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2 n \\times 2 n$ board so that there exists a unique partition of the board into $1 \\times 2$ and $2 \\times 1$ dominoes, none of which conta... |
olympiad_bench_boxed | [
{
"content": "Define $P(n)=n^{2}+n+1$. For any positive integers $a$ and $b$, the set\n\n$$\n\\{P(a), P(a+1), P(a+2), \\ldots, P(a+b)\\}\n$$\n\nis said to be fragrant if none of its elements is relatively prime to the product of the other elements. Determine the smallest size of a fragrant set. Let's think step... | MATH | {
"ground_truth": [
"6"
],
"style": "rule"
} | {
"index": 51,
"raw_problem": "Define $P(n)=n^{2}+n+1$. For any positive integers $a$ and $b$, the set\n\n$$\n\\{P(a), P(a+1), P(a+2), \\ldots, P(a+b)\\}\n$$\n\nis said to be fragrant if none of its elements is relatively prime to the product of the other elements. Determine the smallest size of a fragrant set.",
... |
olympiad_bench_boxed | [
{
"content": "Denote by $\\mathbb{N}$ the set of all positive integers. Find all functions $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-m n$ is nonzero and divides $m f(m)+n f(n)$. Let's think step by step and output the final answer within \\... | MATH | {
"ground_truth": [
"$f(n)=n^{2}$"
],
"style": "rule"
} | {
"index": 52,
"raw_problem": "Denote by $\\mathbb{N}$ the set of all positive integers. Find all functions $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-m n$ is nonzero and divides $m f(m)+n f(n)$.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,\n\n$$\n\\sqrt[N]{\\frac{x^{2 N}+1}{2}} \\leqslant a_{n}(x-1)^{2}+x\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$\\frac{N}{2}$"
],
"style": "rule"
} | {
"index": 53,
"raw_problem": "Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,\n\n$$\n\\sqrt[N]{\\frac{x^{2 N}+1}{2}} \\leqslant a_{n}(x-1)^{2}+x\n$$",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Let $\\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\\mathcal{B}$ denote the subset of $\\mathcal{A}$ formed by all polynomials which can be expressed as\n\n$$\n(x+y+z) P(x, y, z)+(x y+y z+z x) Q(x, y, z)+x y z R(x, y, z)\n$$\n\nwith $P... | MATH | {
"ground_truth": [
"4"
],
"style": "rule"
} | {
"index": 54,
"raw_problem": "Let $\\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\\mathcal{B}$ denote the subset of $\\mathcal{A}$ formed by all polynomials which can be expressed as\n\n$$\n(x+y+z) P(x, y, z)+(x y+y z+z x) Q(x, y, z)+x y z R(x, y, z)\n... |
olympiad_bench_boxed | [
{
"content": "Suppose that $a, b, c, d$ are positive real numbers satisfying $(a+c)(b+d)=a c+b d$. Find the smallest possible value of\n\n$$\nS=\\frac{a}{b}+\\frac{b}{c}+\\frac{c}{d}+\\frac{d}{a}\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"8"
],
"style": "rule"
} | {
"index": 55,
"raw_problem": "Suppose that $a, b, c, d$ are positive real numbers satisfying $(a+c)(b+d)=a c+b d$. Find the smallest possible value of\n\n$$\nS=\\frac{a}{b}+\\frac{b}{c}+\\frac{c}{d}+\\frac{d}{a}\n$$",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Let $\\mathbb{R}^{+}$be the set of positive real numbers. Determine all functions $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$ such that, for all positive real numbers $x$ and $y$,\n\n$$\nf(x+f(x y))+y=f(x) f(y)+1\n\\tag{*}\n$$ Let's think step by step and output the final answer within \\box... | MATH | {
"ground_truth": [
"$f(x)=x+1$"
],
"style": "rule"
} | {
"index": 56,
"raw_problem": "Let $\\mathbb{R}^{+}$be the set of positive real numbers. Determine all functions $f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+}$ such that, for all positive real numbers $x$ and $y$,\n\n$$\nf(x+f(x y))+y=f(x) f(y)+1\n\\tag{*}\n$$",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Let $n$ be an integer with $n \\geqslant 2$. On a slope of a mountain, $n^{2}$ checkpoints are marked, numbered from 1 to $n^{2}$ from the bottom to the top. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars numbered from 1 to $k$; each cable car provides a transfer from some c... | MATH | {
"ground_truth": [
"$n^{2}-n+1$"
],
"style": "rule"
} | {
"index": 57,
"raw_problem": "Let $n$ be an integer with $n \\geqslant 2$. On a slope of a mountain, $n^{2}$ checkpoints are marked, numbered from 1 to $n^{2}$ from the bottom to the top. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars numbered from 1 to $k$; each cable car provides a transfe... |
olympiad_bench_boxed | [
{
"content": "The Fibonacci numbers $F_{0}, F_{1}, F_{2}, \\ldots$ are defined inductively by $F_{0}=0, F_{1}=1$, and $F_{n+1}=F_{n}+F_{n-1}$ for $n \\geqslant 1$. Given an integer $n \\geqslant 2$, determine the smallest size of a set $S$ of integers such that for every $k=2,3, \\ldots, n$ there exist some $x,... | MATH | {
"ground_truth": [
"$\\lceil n / 2\\rceil+1$"
],
"style": "rule"
} | {
"index": 58,
"raw_problem": "The Fibonacci numbers $F_{0}, F_{1}, F_{2}, \\ldots$ are defined inductively by $F_{0}=0, F_{1}=1$, and $F_{n+1}=F_{n}+F_{n-1}$ for $n \\geqslant 1$. Given an integer $n \\geqslant 2$, determine the smallest size of a set $S$ of integers such that for every $k=2,3, \\ldots, n$ there e... |
olympiad_bench_boxed | [
{
"content": "Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1. In every round, player $A$ erases two numbers $x$ and $y$ from the blackboard, and then player $B$ writes one of the numbers $x+y$ and $|x-y|$ on the blackboard. The game terminates as soon as, at ... | MATH | {
"ground_truth": [
"7"
],
"style": "rule"
} | {
"index": 59,
"raw_problem": "Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1. In every round, player $A$ erases two numbers $x$ and $y$ from the blackboard, and then player $B$ writes one of the numbers $x+y$ and $|x-y|$ on the blackboard. The game terminates as... |
olympiad_bench_boxed | [
{
"content": "Let $n$ be a positive integer. Harry has $n$ coins lined up on his desk, each showing heads or tails. He repeatedly does the following operation: if there are $k$ coins showing heads and $k>0$, then he flips the $k^{\\text {th }}$ coin over; otherwise he stops the process. (For example, the proces... | MATH | {
"ground_truth": [
"$\\frac{1}{4} n(n+1)$"
],
"style": "rule"
} | {
"index": 60,
"raw_problem": "Let $n$ be a positive integer. Harry has $n$ coins lined up on his desk, each showing heads or tails. He repeatedly does the following operation: if there are $k$ coins showing heads and $k>0$, then he flips the $k^{\\text {th }}$ coin over; otherwise he stops the process. (For exampl... |
olympiad_bench_boxed | [
{
"content": "On a flat plane in Camelot, King Arthur builds a labyrinth $\\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue.... | MATH | {
"ground_truth": [
"$k=n+1$"
],
"style": "rule"
} | {
"index": 61,
"raw_problem": "On a flat plane in Camelot, King Arthur builds a labyrinth $\\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side e... |
olympiad_bench_boxed | [
{
"content": "There are 60 empty boxes $B_{1}, \\ldots, B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.\n\nIn the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round ... | MATH | {
"ground_truth": [
"960"
],
"style": "rule"
} | {
"index": 62,
"raw_problem": "There are 60 empty boxes $B_{1}, \\ldots, B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.\n\nIn the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subs... |
olympiad_bench_boxed | [
{
"content": "For any two different real numbers $x$ and $y$, we define $D(x, y)$ to be the unique integer $d$ satisfying $2^{d} \\leqslant|x-y|<2^{d+1}$. Given a set of reals $\\mathcal{F}$, and an element $x \\in \\mathcal{F}$, we say that the scales of $x$ in $\\mathcal{F}$ are the values of $D(x, y)$ for $y... | MATH | {
"ground_truth": [
"$2^{k}$"
],
"style": "rule"
} | {
"index": 63,
"raw_problem": "For any two different real numbers $x$ and $y$, we define $D(x, y)$ to be the unique integer $d$ satisfying $2^{d} \\leqslant|x-y|<2^{d+1}$. Given a set of reals $\\mathcal{F}$, and an element $x \\in \\mathcal{F}$, we say that the scales of $x$ in $\\mathcal{F}$ are the values of $D(... |
olympiad_bench_boxed | [
{
"content": "Find all pairs $(m, n)$ of positive integers satisfying the equation\n\n$$\n\\left(2^{n}-1\\right)\\left(2^{n}-2\\right)\\left(2^{n}-4\\right) \\cdots\\left(2^{n}-2^{n-1}\\right)=m !\n\\tag{1}\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$(1,1), (3,2)$"
],
"style": "rule"
} | {
"index": 64,
"raw_problem": "Find all pairs $(m, n)$ of positive integers satisfying the equation\n\n$$\n\\left(2^{n}-1\\right)\\left(2^{n}-2\\right)\\left(2^{n}-4\\right) \\cdots\\left(2^{n}-2^{n-1}\\right)=m !\n\\tag{1}\n$$",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)$"
],
"style": "rule"
} | {
"index": 65,
"raw_problem": "Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Determine all functions $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ with the property that\n\n$$\nf(x-f(y))=f(f(x))-f(y)-1\n\\tag{1}\n$$\n\nholds for all $x, y \\in \\mathbb{Z}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$f(x)=-1$,$f(x)=x+1$"
],
"style": "rule"
} | {
"index": 66,
"raw_problem": "Determine all functions $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ with the property that\n\n$$\nf(x-f(y))=f(f(x))-f(y)-1\n\\tag{1}\n$$\n\nholds for all $x, y \\in \\mathbb{Z}$.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Let $n$ be a fixed positive integer. Find the maximum possible value of\n\n$$\n\\sum_{1 \\leqslant r<s \\leqslant 2 n}(s-r-n) x_{r} x_{s}\n$$\n\nwhere $-1 \\leqslant x_{i} \\leqslant 1$ for all $i=1,2, \\ldots, 2 n$. Let's think step by step and output the final answer within \\boxed{}.",
"rol... | MATH | {
"ground_truth": [
"$n(n-1)$"
],
"style": "rule"
} | {
"index": 67,
"raw_problem": "Let $n$ be a fixed positive integer. Find the maximum possible value of\n\n$$\n\\sum_{1 \\leqslant r<s \\leqslant 2 n}(s-r-n) x_{r} x_{s}\n$$\n\nwhere $-1 \\leqslant x_{i} \\leqslant 1$ for all $i=1,2, \\ldots, 2 n$.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Find all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ satisfying the equation\n\n$$\nf(x+f(x+y))+f(x y)=x+f(x+y)+y f(x)\\tag{1}\n$$\n\nfor all real numbers $x$ and $y$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$f(x)=x$,$f(x)=2-x$"
],
"style": "rule"
} | {
"index": 68,
"raw_problem": "Find all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ satisfying the equation\n\n$$\nf(x+f(x+y))+f(x y)=x+f(x+y)+y f(x)\\tag{1}\n$$\n\nfor all real numbers $x$ and $y$.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "For a finite set $A$ of positive integers, we call a partition of $A$ into two disjoint nonempty subsets $A_{1}$ and $A_{2}$ good if the least common multiple of the elements in $A_{1}$ is equal to the greatest common divisor of the elements in $A_{2}$. Determine the minimum value of $n$ such that... | MATH | {
"ground_truth": [
"3024"
],
"style": "rule"
} | {
"index": 69,
"raw_problem": "For a finite set $A$ of positive integers, we call a partition of $A$ into two disjoint nonempty subsets $A_{1}$ and $A_{2}$ good if the least common multiple of the elements in $A_{1}$ is equal to the greatest common divisor of the elements in $A_{2}$. Determine the minimum value of ... |
olympiad_bench_boxed | [
{
"content": "Let $A B C$ be an acute triangle, and let $M$ be the midpoint of $A C$. A circle $\\omega$ passing through $B$ and $M$ meets the sides $A B$ and $B C$ again at $P$ and $Q$, respectively. Let $T$ be the point such that the quadrilateral $B P T Q$ is a parallelogram. Suppose that $T$ lies on the cir... | MATH | {
"ground_truth": [
"$\\sqrt{2}$"
],
"style": "rule"
} | {
"index": 70,
"raw_problem": "Let $A B C$ be an acute triangle, and let $M$ be the midpoint of $A C$. A circle $\\omega$ passing through $B$ and $M$ meets the sides $A B$ and $B C$ again at $P$ and $Q$, respectively. Let $T$ be the point such that the quadrilateral $B P T Q$ is a parallelogram. Suppose that $T$ li... |
olympiad_bench_boxed | [
{
"content": "Determine all triples $(a, b, c)$ of positive integers for which $a b-c, b c-a$, and $c a-b$ are powers of 2 .\n\nExplanation: A power of 2 is an integer of the form $2^{n}$, where $n$ denotes some nonnegative integer. Let's think step by step and output the final answer within \\boxed{}.",
"r... | MATH | {
"ground_truth": [
"$(2,2,2),(2,2,3),(2,3,2),(3,2,2),(2,6,11),(2,11,6),(6,2,11),(6,11,2),(11,2,6),(11,6,2),(3,5,7),(3,7,5),(5,3,7),(5,7,3),(7,3,5),(7,5,3)$"
],
"style": "rule"
} | {
"index": 71,
"raw_problem": "Determine all triples $(a, b, c)$ of positive integers for which $a b-c, b c-a$, and $c a-b$ are powers of 2 .\n\nExplanation: A power of 2 is an integer of the form $2^{n}$, where $n$ denotes some nonnegative integer.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Let $\\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \\mathbb{Z}_{>0} \\rightarrow \\mathbb{Z}_{>0}$ is called $k$-good if $\\operatorname{gcd}(f(m)+n, f(n)+m) \\leqslant k$ for all $m \\neq n$. Find all $k$ such that there exists a $k$-good func... | MATH | {
"ground_truth": [
"$k \\geqslant 2$"
],
"style": "rule"
} | {
"index": 72,
"raw_problem": "Let $\\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \\mathbb{Z}_{>0} \\rightarrow \\mathbb{Z}_{>0}$ is called $k$-good if $\\operatorname{gcd}(f(m)+n, f(n)+m) \\leqslant k$ for all $m \\neq n$. Find all $k$ such that there exists a ... |
olympiad_bench_boxed | [
{
"content": "For a sequence $x_{1}, x_{2}, \\ldots, x_{n}$ of real numbers, we define its price as\n\n$$\n\\max _{1 \\leqslant i \\leqslant n}\\left|x_{1}+\\cdots+x_{i}\\right|\n$$\n\nGiven $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possibl... | MATH | {
"ground_truth": [
"2"
],
"style": "rule"
} | {
"index": 73,
"raw_problem": "For a sequence $x_{1}, x_{2}, \\ldots, x_{n}$ of real numbers, we define its price as\n\n$$\n\\max _{1 \\leqslant i \\leqslant n}\\left|x_{1}+\\cdots+x_{i}\\right|\n$$\n\nGiven $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave check... |
olympiad_bench_boxed | [
{
"content": "Determine all functions $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ satisfying\n\n$$\nf(f(m)+n)+f(m)=f(n)+f(3 m)+2014\n\\tag{1}\n$$\n\nfor all integers $m$ and $n$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$f(n) = 2 n+1007$"
],
"style": "rule"
} | {
"index": 74,
"raw_problem": "Determine all functions $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ satisfying\n\n$$\nf(f(m)+n)+f(m)=f(n)+f(3 m)+2014\n\\tag{1}\n$$\n\nfor all integers $m$ and $n$.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has\n\n$$\n\\left|y^{2}-P(x)\\right| \\leqslant 2|x| \\text { if and only if }\\left|x^{2}-P(y)\\right| \\leqslant 2|y|\n\\tag{1}\n$$\n\nDetermine all possible values o... | MATH | {
"ground_truth": [
"$(-\\infty, 0) \\cup\\{1\\}$."
],
"style": "rule"
} | {
"index": 75,
"raw_problem": "Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has\n\n$$\n\\left|y^{2}-P(x)\\right| \\leqslant 2|x| \\text { if and only if }\\left|x^{2}-P(y)\\right| \\leqslant 2|y|\n\\tag{1}\n$$\n\nDetermine all poss... |
olympiad_bench_boxed | [
{
"content": "Let $n \\geqslant 2$ be an integer. Consider an $n \\times n$ chessboard divided into $n^{2}$ unit squares. We call a configuration of $n$ rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that for every happy configurati... | MATH | {
"ground_truth": [
"$\\lfloor\\sqrt{n-1}\\rfloor$"
],
"style": "rule"
} | {
"index": 76,
"raw_problem": "Let $n \\geqslant 2$ be an integer. Consider an $n \\times n$ chessboard divided into $n^{2}$ unit squares. We call a configuration of $n$ rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that for every happ... |
olympiad_bench_boxed | [
{
"content": "We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of 100 cards each from this deck. We would like to define a rule that declares one of t... | MATH | {
"ground_truth": [
"100"
],
"style": "rule"
} | {
"index": 77,
"raw_problem": "We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of 100 cards each from this deck. We would like to define a rule that decl... |
olympiad_bench_boxed | [
{
"content": "Let $n \\geqslant 2$ be an integer, and let $A_{n}$ be the set\n\n$$\nA_{n}=\\left\\{2^{n}-2^{k} \\mid k \\in \\mathbb{Z}, 0 \\leqslant k<n\\right\\} .\n$$\n\nDetermine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_{n}$. Let... | MATH | {
"ground_truth": [
"$(n-2) 2^{n}+1$"
],
"style": "rule"
} | {
"index": 78,
"raw_problem": "Let $n \\geqslant 2$ be an integer, and let $A_{n}$ be the set\n\n$$\nA_{n}=\\left\\{2^{n}-2^{k} \\mid k \\in \\mathbb{Z}, 0 \\leqslant k<n\\right\\} .\n$$\n\nDetermine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of... |
olympiad_bench_boxed | [
{
"content": "Let $k \\geqslant 2$ be an integer. Find the smallest integer $n \\geqslant k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set. Let's think step by step and output the final an... | MATH | {
"ground_truth": [
"$n=k+4$"
],
"style": "rule"
} | {
"index": 79,
"raw_problem": "Let $k \\geqslant 2$ be an integer. Find the smallest integer $n \\geqslant k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Let $\\mathbb{R}_{>0}$ be the set of positive real numbers. Find all functions $f: \\mathbb{R}_{>0} \\rightarrow \\mathbb{R}_{>0}$ such that, for every $x \\in \\mathbb{R}_{>0}$, there exists a unique $y \\in \\mathbb{R}_{>0}$ satisfying\n\n$$\nx f(y)+y f(x) \\leqslant 2 .\n$$ Let's think step by ... | MATH | {
"ground_truth": [
"$f(x)=\\frac{1}{x}$"
],
"style": "rule"
} | {
"index": 80,
"raw_problem": "Let $\\mathbb{R}_{>0}$ be the set of positive real numbers. Find all functions $f: \\mathbb{R}_{>0} \\rightarrow \\mathbb{R}_{>0}$ such that, for every $x \\in \\mathbb{R}_{>0}$, there exists a unique $y \\in \\mathbb{R}_{>0}$ satisfying\n\n$$\nx f(y)+y f(x) \\leqslant 2 .\n$$",
"sp... |
olympiad_bench_boxed | [
{
"content": "Find all positive integers $n \\geqslant 2$ for which there exist $n$ real numbers $a_{1}<\\cdots<a_{n}$ and a real number $r>0$ such that the $\\frac{1}{2} n(n-1)$ differences $a_{j}-a_{i}$ for $1 \\leqslant i<j \\leqslant n$ are equal, in some order, to the numbers $r^{1}, r^{2}, \\ldots, r^{\\f... | MATH | {
"ground_truth": [
"2,3,4"
],
"style": "rule"
} | {
"index": 81,
"raw_problem": "Find all positive integers $n \\geqslant 2$ for which there exist $n$ real numbers $a_{1}<\\cdots<a_{n}$ and a real number $r>0$ such that the $\\frac{1}{2} n(n-1)$ differences $a_{j}-a_{i}$ for $1 \\leqslant i<j \\leqslant n$ are equal, in some order, to the numbers $r^{1}, r^{2}, \\... |
olympiad_bench_boxed | [
{
"content": "$A \\pm 1 \\text{-}sequence$ is a sequence of 2022 numbers $a_{1}, \\ldots, a_{2022}$, each equal to either +1 or -1 . Determine the largest $C$ so that, for any $\\pm 1 -sequence$, there exists an integer $k$ and indices $1 \\leqslant t_{1}<\\ldots<t_{k} \\leqslant 2022$ so that $t_{i+1}-t_{i} \\... | MATH | {
"ground_truth": [
"506"
],
"style": "rule"
} | {
"index": 82,
"raw_problem": "$A \\pm 1 \\text{-}sequence$ is a sequence of 2022 numbers $a_{1}, \\ldots, a_{2022}$, each equal to either +1 or -1 . Determine the largest $C$ so that, for any $\\pm 1 -sequence$, there exists an integer $k$ and indices $1 \\leqslant t_{1}<\\ldots<t_{k} \\leqslant 2022$ so that $t_{... |
olympiad_bench_boxed | [
{
"content": "In each square of a garden shaped like a $2022 \\times 2022$ board, there is initially a tree of height 0 . A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:\n\n- The gardener chooses a square in the garden. Each tree on that square an... | MATH | {
"ground_truth": [
"2271380"
],
"style": "rule"
} | {
"index": 83,
"raw_problem": "In each square of a garden shaped like a $2022 \\times 2022$ board, there is initially a tree of height 0 . A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:\n\n- The gardener chooses a square in the garden. Each tree on t... |
olympiad_bench_boxed | [
{
"content": "Lucy starts by writing $s$ integer-valued 2022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\\mathbf{v}=\\left(v_{1}, \\ldots, v_{2022}\\right)$ and $\\mathbf{w}=\\left(w_{1}, \\ldots, w_{2022}\\right)$ that she has already written, and apply on... | MATH | {
"ground_truth": [
"3"
],
"style": "rule"
} | {
"index": 84,
"raw_problem": "Lucy starts by writing $s$ integer-valued 2022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\\mathbf{v}=\\left(v_{1}, \\ldots, v_{2022}\\right)$ and $\\mathbf{w}=\\left(w_{1}, \\ldots, w_{2022}\\right)$ that she has already written,... |
olympiad_bench_boxed | [
{
"content": "Alice fills the fields of an $n \\times n$ board with numbers from 1 to $n^{2}$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such that:\n\n(i) The first field in the sequence... | MATH | {
"ground_truth": [
"$2 n^{2}-2 n+1$"
],
"style": "rule"
} | {
"index": 85,
"raw_problem": "Alice fills the fields of an $n \\times n$ board with numbers from 1 to $n^{2}$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such that:\n\n(i) The first field in... |
olympiad_bench_boxed | [
{
"content": "Let $\\mathbb{Z}_{\\geqslant 0}$ be the set of non-negative integers, and let $f: \\mathbb{Z}_{\\geqslant 0} \\times \\mathbb{Z}_{\\geqslant 0} \\rightarrow \\mathbb{Z}_{\\geqslant 0}$ be a bijection such that whenever $f\\left(x_{1}, y_{1}\\right)>f\\left(x_{2}, y_{2}\\right)$, we have $f\\left(x... | MATH | {
"ground_truth": [
"2500,7500"
],
"style": "rule"
} | {
"index": 86,
"raw_problem": "Let $\\mathbb{Z}_{\\geqslant 0}$ be the set of non-negative integers, and let $f: \\mathbb{Z}_{\\geqslant 0} \\times \\mathbb{Z}_{\\geqslant 0} \\rightarrow \\mathbb{Z}_{\\geqslant 0}$ be a bijection such that whenever $f\\left(x_{1}, y_{1}\\right)>f\\left(x_{2}, y_{2}\\right)$, we ha... |
olympiad_bench_boxed | [
{
"content": "A number is called Norwegian if it has three distinct positive divisors whose sum is equal to 2022. Determine the smallest Norwegian number.\n\n(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.) Let's think step by step and output the final answer w... | MATH | {
"ground_truth": [
"1344"
],
"style": "rule"
} | {
"index": 87,
"raw_problem": "A number is called Norwegian if it has three distinct positive divisors whose sum is equal to 2022. Determine the smallest Norwegian number.\n\n(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.)",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Find all positive integers $n>2$ such that\n\n$$\nn ! \\mid \\prod_{\\substack{p<q \\leqslant n \\\\ p, q \\text { primes }}}(p+q)\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"7"
],
"style": "rule"
} | {
"index": 88,
"raw_problem": "Find all positive integers $n>2$ such that\n\n$$\nn ! \\mid \\prod_{\\substack{p<q \\leqslant n \\\\ p, q \\text { primes }}}(p+q)\n$$",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Find all triples of positive integers $(a, b, p)$ with $p$ prime and\n\n$$\na^{p}=b !+p\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | MATH | {
"ground_truth": [
"$(2,2,2),(3,4,3)$"
],
"style": "rule"
} | {
"index": 89,
"raw_problem": "Find all triples of positive integers $(a, b, p)$ with $p$ prime and\n\n$$\na^{p}=b !+p\n$$",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Let $\\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f: \\mathbb{Q}_{>0} \\rightarrow \\mathbb{Q}_{>0}$ satisfying\n\n$$\nf\\left(x^{2} f(y)^{2}\\right)=f(x)^{2} f(y)\n\\tag{*}\n$$\n\nfor all $x, y \\in \\mathbb{Q}_{>0}$. Let's think step by step and out... | MATH | {
"ground_truth": [
"$f(x)=1$"
],
"style": "rule"
} | {
"index": 90,
"raw_problem": "Let $\\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f: \\mathbb{Q}_{>0} \\rightarrow \\mathbb{Q}_{>0}$ satisfying\n\n$$\nf\\left(x^{2} f(y)^{2}\\right)=f(x)^{2} f(y)\n\\tag{*}\n$$\n\nfor all $x, y \\in \\mathbb{Q}_{>0}$.",
"split": "test"... |
olympiad_bench_boxed | [
{
"content": "Let $a_{0}, a_{1}, a_{2}, \\ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \\geqslant 2$ there exists $1 \\leqslant k \\leqslant n$ satisfying\n\n$$\na_{n}=\\frac{a_{n-1}+\\cdots+a_{n-k}}{k}\n$$\n\nFind the maximal possible value of $a_{2018}-a_{2017}$. Let's t... | MATH | {
"ground_truth": [
"$\\frac{2016}{2017^{2}}$"
],
"style": "rule"
} | {
"index": 91,
"raw_problem": "Let $a_{0}, a_{1}, a_{2}, \\ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \\geqslant 2$ there exists $1 \\leqslant k \\leqslant n$ satisfying\n\n$$\na_{n}=\\frac{a_{n-1}+\\cdots+a_{n-k}}{k}\n$$\n\nFind the maximal possible value of $a_{2018}-a_{20... |
olympiad_bench_boxed | [
{
"content": "Find the maximal value of\n\n$$\nS=\\sqrt[3]{\\frac{a}{b+7}}+\\sqrt[3]{\\frac{b}{c+7}}+\\sqrt[3]{\\frac{c}{d+7}}+\\sqrt[3]{\\frac{d}{a+7}}\n$$\n\nwhere $a, b, c, d$ are nonnegative real numbers which satisfy $a+b+c+d=100$. Let's think step by step and output the final answer within \\boxed{}.",
... | MATH | {
"ground_truth": [
"$\\frac{8}{\\sqrt[3]{7}}$"
],
"style": "rule"
} | {
"index": 92,
"raw_problem": "Find the maximal value of\n\n$$\nS=\\sqrt[3]{\\frac{a}{b+7}}+\\sqrt[3]{\\frac{b}{c+7}}+\\sqrt[3]{\\frac{c}{d+7}}+\\sqrt[3]{\\frac{d}{a+7}}\n$$\n\nwhere $a, b, c, d$ are nonnegative real numbers which satisfy $a+b+c+d=100$.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Queenie and Horst play a game on a $20 \\times 20$ chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game g... | MATH | {
"ground_truth": [
"100"
],
"style": "rule"
} | {
"index": 93,
"raw_problem": "Queenie and Horst play a game on a $20 \\times 20$ chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty squar... |
olympiad_bench_boxed | [
{
"content": "Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of h... | MATH | {
"ground_truth": [
"$k\\left(4 k^{2}+k-1\\right) / 2$"
],
"style": "rule"
} | {
"index": 94,
"raw_problem": "Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs... |
olympiad_bench_boxed | [
{
"content": "A circle $\\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold:\n\n(i) each triangle from $T$ is inscribed in $\\omega$;\n\n(ii) no two triangles from $T$ have a common interior point.\n\nDetermine all positive real numbers $t$ such that, fo... | MATH | {
"ground_truth": [
"$t(0,4]$"
],
"style": "rule"
} | {
"index": 95,
"raw_problem": "A circle $\\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold:\n\n(i) each triangle from $T$ is inscribed in $\\omega$;\n\n(ii) no two triangles from $T$ have a common interior point.\n\nDetermine all positive real numbers $t$ ... |
olympiad_bench_boxed | [
{
"content": "Let $n$ be a positive integer. Find the smallest integer $k$ with the following property: Given any real numbers $a_{1}, \\ldots, a_{d}$ such that $a_{1}+a_{2}+\\cdots+a_{d}=n$ and $0 \\leqslant a_{i} \\leqslant 1$ for $i=1,2, \\ldots, d$, it is possible to partition these numbers into $k$ groups ... | MATH | {
"ground_truth": [
"$k=2 n-1$"
],
"style": "rule"
} | {
"index": 96,
"raw_problem": "Let $n$ be a positive integer. Find the smallest integer $k$ with the following property: Given any real numbers $a_{1}, \\ldots, a_{d}$ such that $a_{1}+a_{2}+\\cdots+a_{d}=n$ and $0 \\leqslant a_{i} \\leqslant 1$ for $i=1,2, \\ldots, d$, it is possible to partition these numbers int... |
olympiad_bench_boxed | [
{
"content": "In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw $k$ lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of bot... | MATH | {
"ground_truth": [
"2013"
],
"style": "rule"
} | {
"index": 97,
"raw_problem": "In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw $k$ lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains ... |
olympiad_bench_boxed | [
{
"content": "Let $\\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \\mathbb{Z}_{>0} \\rightarrow \\mathbb{Z}_{>0}$ such that\n\n$$\nm^{2}+f(n) \\mid m f(m)+n\n$$\n\nfor all positive integers $m$ and $n$. Let's think step by step and output the final answer within \\boxed{}.",
"role... | MATH | {
"ground_truth": [
"$f(n)=n$"
],
"style": "rule"
} | {
"index": 98,
"raw_problem": "Let $\\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \\mathbb{Z}_{>0} \\rightarrow \\mathbb{Z}_{>0}$ such that\n\n$$\nm^{2}+f(n) \\mid m f(m)+n\n$$\n\nfor all positive integers $m$ and $n$.",
"split": "test"
} |
olympiad_bench_boxed | [
{
"content": "Find the largest possible integer $k$, such that the following statement is true:\n\nLet 2009 arbitrary non-degenerated triangles be given. In every triangle the three sides are colored, such that one is blue, one is red and one is white. Now, for every color separately, let us sort the lengths of... | MATH | {
"ground_truth": [
"1"
],
"style": "rule"
} | {
"index": 99,
"raw_problem": "Find the largest possible integer $k$, such that the following statement is true:\n\nLet 2009 arbitrary non-degenerated triangles be given. In every triangle the three sides are colored, such that one is blue, one is red and one is white. Now, for every color separately, let us sort t... |
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