data_source stringclasses 1
value | answer stringlengths 1 204 | source_dataset stringclasses 4
values | prompt listlengths 2 2 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|---|
numina_math | $M=\frac{9}{32} \sqrt{2}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the smallest number $M$ such that the inequality\n\n$$\n\\left|a b\\left(a^{2}-b^{2}\\right)+b c\\left(b^{2}-c^{2}\\right)+c a\\left(c^{2}-a^{2}\\right)\\right| \\leq M... | math | {
"ground_truth": "$M=\\frac{9}{32} \\sqrt{2}$",
"style": "rule"
} | {
"index": 800,
"split": "test"
} |
numina_math | $1003$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagon... | math | {
"ground_truth": "$1003$",
"style": "rule"
} | {
"index": 801,
"split": "test"
} |
numina_math | $\angle B E A_{1}=90$,$\angle A E B_{1}=90$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In triangle $A B C$, let $J$ be the centre of the excircle tangent to side $B C$ at $A_{1}$ and to the extensions of sides $A C$ and $A B$ at $B_{1}$ and $C_{1}$, respectively. S... | math | {
"ground_truth": "$\\angle B E A_{1}=90$,$\\angle A E B_{1}=90$",
"style": "rule"
} | {
"index": 802,
"split": "test"
} |
numina_math | $(0,2),(0,-2),(4,23),(4,-23)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all pairs $(x, y)$ of integers satisfying the equation\n\n$$\n1+2^{x}+2^{2 x+1}=y^{2}\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"... | math | {
"ground_truth": "$(0,2),(0,-2),(4,23),(4,-23)$",
"style": "rule"
} | {
"index": 803,
"split": "test"
} |
numina_math | $\left\lfloor\log _{2} n\right\rfloor+1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | math | {
"ground_truth": "$\\left\\lfloor\\log _{2} n\\right\\rfloor+1$",
"style": "rule"
} | {
"index": 804,
"split": "test"
} |
numina_math | $m_{\max }=n^{2}-n-1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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}$,... | math | {
"ground_truth": "$m_{\\max }=n^{2}-n-1$",
"style": "rule"
} | {
"index": 805,
"split": "test"
} |
numina_math | $\frac{100!}{2^{50}}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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,... | math | {
"ground_truth": "$\\frac{100!}{2^{50}}$",
"style": "rule"
} | {
"index": 806,
"split": "test"
} |
numina_math | 2 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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}{... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 807,
"split": "test"
} |
numina_math | 1,3 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | math | {
"ground_truth": "1,3",
"style": "rule"
} | {
"index": 808,
"split": "test"
} |
numina_math | $-2,0,2$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 f... | math | {
"ground_truth": "$-2,0,2$",
"style": "rule"
} | {
"index": 809,
"split": "test"
} |
numina_math | $-(n-1) / 2$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $y_{1}, y_{2}, \\ldots, y_{... | math | {
"ground_truth": "$-(n-1) / 2$",
"style": "rule"
} | {
"index": 810,
"split": "test"
} |
numina_math | $\frac{n(n+1)(2 n+1)}{6}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 consist... | math | {
"ground_truth": "$\\frac{n(n+1)(2 n+1)}{6}$",
"style": "rule"
} | {
"index": 811,
"split": "test"
} |
numina_math | $n^{2}+1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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. T... | math | {
"ground_truth": "$n^{2}+1$",
"style": "rule"
} | {
"index": 812,
"split": "test"
} |
numina_math | 6048 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 co... | math | {
"ground_truth": "6048",
"style": "rule"
} | {
"index": 813,
"split": "test"
} |
numina_math | 807 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | math | {
"ground_truth": "807",
"style": "rule"
} | {
"index": 814,
"split": "test"
} |
numina_math | $(3,2)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | math | {
"ground_truth": "$(3,2)$",
"style": "rule"
} | {
"index": 815,
"split": "test"
} |
numina_math | 3 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 816,
"split": "test"
} |
numina_math | $\frac{1}{2}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 su... | math | {
"ground_truth": "$\\frac{1}{2}$",
"style": "rule"
} | {
"index": 817,
"split": "test"
} |
numina_math | 2016 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | math | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": 818,
"split": "test"
} |
numina_math | $\frac{4}{9}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $0=x_{0}<x_{1}<x_{2}<\\cdots<x_{n}$, w... | math | {
"ground_truth": "$\\frac{4}{9}$",
"style": "rule"
} | {
"index": 819,
"split": "test"
} |
numina_math | 1 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 820,
"split": "test"
} |
numina_math | $2 n$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ther... | math | {
"ground_truth": "$2 n$",
"style": "rule"
} | {
"index": 821,
"split": "test"
} |
numina_math | 6 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 i... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 822,
"split": "test"
} |
numina_math | $f(n)=n^{2}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 intege... | math | {
"ground_truth": "$f(n)=n^{2}$",
"style": "rule"
} | {
"index": 823,
"split": "test"
} |
numina_math | $\frac{N}{2}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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$$\n\\sqrt[N]{\\frac{x^{2 N}+1}{2}} \\leqslant a_{n}(... | math | {
"ground_truth": "$\\frac{N}{2}$",
"style": "rule"
} | {
"index": 824,
"split": "test"
} |
numina_math | 4 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\\mathcal{B}$ denote the subset of $\\mathcal{A}$ formed by all ... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 825,
"split": "test"
} |
numina_math | 8 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $a, b, c, d$ are positive real numbers satisfying $(a+c)(b+d)=a c+b d$. Find the smallest possible value of\n\n$$\nS=\\frac{a}{b}+\\frac{b}{c}+\\frac{c}{d}+\\frac{d}... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 826,
"split": "test"
} |
numina_math | $f(x)=x+1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $x$ a... | math | {
"ground_truth": "$f(x)=x+1$",
"style": "rule"
} | {
"index": 827,
"split": "test"
} |
numina_math | $n^{2}-n+1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ca... | math | {
"ground_truth": "$n^{2}-n+1$",
"style": "rule"
} | {
"index": 828,
"split": "test"
} |
numina_math | $\lceil n / 2\rceil+1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $n \\geqsla... | math | {
"ground_truth": "$\\lceil n / 2\\rceil+1$",
"style": "rule"
} | {
"index": 829,
"split": "test"
} |
numina_math | 7 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 830,
"split": "test"
} |
numina_math | $\frac{1}{4} n(n+1)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 h... | math | {
"ground_truth": "$\\frac{1}{4} n(n+1)$",
"style": "rule"
} | {
"index": 831,
"split": "test"
} |
numina_math | $k=n+1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | math | {
"ground_truth": "$k=n+1$",
"style": "rule"
} | {
"index": 832,
"split": "test"
} |
numina_math | 960 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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\n... | math | {
"ground_truth": "960",
"style": "rule"
} | {
"index": 833,
"split": "test"
} |
numina_math | $2^{k}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $\\mathcal{F}$, ... | math | {
"ground_truth": "$2^{k}$",
"style": "rule"
} | {
"index": 834,
"split": "test"
} |
numina_math | $(1,1), (3,2)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all pairs $(m, n)$ of positive integers satisfying the equation\n\n$$\n\\left(2^{n}-1\\right)\\left(2^{n}-2\\right)\\left(2^{n}-4\\right) \\cdots\\left(2^{n}-2^{n-1}\\right)... | math | {
"ground_truth": "$(1,1), (3,2)$",
"style": "rule"
} | {
"index": 835,
"split": "test"
} |
numina_math | $(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": ... | math | {
"ground_truth": "$(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)$",
"style": "rule"
} | {
"index": 836,
"split": "test"
} |
numina_math | $f(x)=-1$,$f(x)=x+1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all functions $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ with the property that\n\n$$\nf(x-f(y))=f(f(x))-f(y)-1\n\\tag{1}\n$$\n\nholds for all $x, y \\in \\mathbb{Z}$. L... | math | {
"ground_truth": "$f(x)=-1$,$f(x)=x+1$",
"style": "rule"
} | {
"index": 837,
"split": "test"
} |
numina_math | $n(n-1)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ be a fixed positive integer. Find the maximum possible value of\n\n$$\n\\sum_{1 \\leqslant r<s \\leqslant 2 n}(s-r-n) x_{r} x_{s}\n$$\n\nwhere $-1 \\leqslant x_{i} \\leqs... | math | {
"ground_truth": "$n(n-1)$",
"style": "rule"
} | {
"index": 838,
"split": "test"
} |
numina_math | $f(x)=x$,$f(x)=2-x$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ satisfying the equation\n\n$$\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$. Le... | math | {
"ground_truth": "$f(x)=x$,$f(x)=2-x$",
"style": "rule"
} | {
"index": 839,
"split": "test"
} |
numina_math | 3024 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | math | {
"ground_truth": "3024",
"style": "rule"
} | {
"index": 840,
"split": "test"
} |
numina_math | $\sqrt{2}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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$, respe... | math | {
"ground_truth": "$\\sqrt{2}$",
"style": "rule"
} | {
"index": 841,
"split": "test"
} |
numina_math | $(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)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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}$, whe... | 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": 842,
"split": "test"
} |
numina_math | $k \geqslant 2$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $\\... | math | {
"ground_truth": "$k \\geqslant 2$",
"style": "rule"
} | {
"index": 843,
"split": "test"
} |
numina_math | $f(x)=a x+b$, where $b$ is an arbitrary integer, and $a$ is an arbitrary positive integer with $\mho(a)=0$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For every positive integer $n$ with prime factorization $n=\\prod_{i=1}^{k} p_{i}^{\\alpha_{i}}$, define\n\n$$\n\\mho(n)=\\sum_{i: p_{i}>10^{100}} \\alpha_{i}\\tag{1}\n$$\n\nThat... | math | {
"ground_truth": "$f(x)=a x+b$, where $b$ is an arbitrary integer, and $a$ is an arbitrary positive integer with $\\mho(a)=0$",
"style": "rule"
} | {
"index": 844,
"split": "test"
} |
numina_math | 2 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For a sequence $x_{1}, x_{2}, \\ldots, x_{n}$ of real numbers, we define its price as\n\n$$\n\\max _{1 \\leqslant i \\leqslant n}\\left|x_{1}+\\cdots+x_{i}\\right|\n$$\n\nGiven $... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 845,
"split": "test"
} |
numina_math | $f(n) = 2 n+1007$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all functions $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ satisfying\n\n$$\nf(f(m)+n)+f(m)=f(n)+f(3 m)+2014\n\\tag{1}\n$$\n\nfor all integers $m$ and $n$. Let's think ste... | math | {
"ground_truth": "$f(n) = 2 n+1007$",
"style": "rule"
} | {
"index": 846,
"split": "test"
} |
numina_math | $(-\infty, 0) \cup\{1\}$. | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has\n\n$$\n\\left|y^{2}-P(x)\\right| \\leqslant ... | math | {
"ground_truth": "$(-\\infty, 0) \\cup\\{1\\}$.",
"style": "rule"
} | {
"index": 847,
"split": "test"
} |
numina_math | $\lfloor\sqrt{n-1}\rfloor$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | math | {
"ground_truth": "$\\lfloor\\sqrt{n-1}\\rfloor$",
"style": "rule"
} | {
"index": 848,
"split": "test"
} |
numina_math | 100 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 player... | math | {
"ground_truth": "100",
"style": "rule"
} | {
"index": 849,
"split": "test"
} |
numina_math | $(n-2) 2^{n}+1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 large... | math | {
"ground_truth": "$(n-2) 2^{n}+1$",
"style": "rule"
} | {
"index": 850,
"split": "test"
} |
numina_math | $n=k+4$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 elem... | math | {
"ground_truth": "$n=k+4$",
"style": "rule"
} | {
"index": 851,
"split": "test"
} |
numina_math | $f(x)=\frac{1}{x}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $x \\in \\mathbb{R}_{>0}$,... | math | {
"ground_truth": "$f(x)=\\frac{1}{x}$",
"style": "rule"
} | {
"index": 852,
"split": "test"
} |
numina_math | 2,3,4 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 $... | math | {
"ground_truth": "2,3,4",
"style": "rule"
} | {
"index": 853,
"split": "test"
} |
numina_math | 506 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "$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 -sequenc... | math | {
"ground_truth": "506",
"style": "rule"
} | {
"index": 854,
"split": "test"
} |
numina_math | 2271380 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | math | {
"ground_truth": "2271380",
"style": "rule"
} | {
"index": 855,
"split": "test"
} |
numina_math | 3 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 856,
"split": "test"
} |
numina_math | $2 n^{2}-2 n+1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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.... | math | {
"ground_truth": "$2 n^{2}-2 n+1$",
"style": "rule"
} | {
"index": 857,
"split": "test"
} |
numina_math | 2500,7500 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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}_{\\geqsla... | math | {
"ground_truth": "2500,7500",
"style": "rule"
} | {
"index": 858,
"split": "test"
} |
numina_math | 1344 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 positiv... | math | {
"ground_truth": "1344",
"style": "rule"
} | {
"index": 859,
"split": "test"
} |
numina_math | 7 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 fin... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 860,
"split": "test"
} |
numina_math | $(2,2,2),(3,4,3)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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": "... | math | {
"ground_truth": "$(2,2,2),(3,4,3)$",
"style": "rule"
} | {
"index": 861,
"split": "test"
} |
numina_math | $f(x)=1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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$$\nf\\left(x^{2... | math | {
"ground_truth": "$f(x)=1$",
"style": "rule"
} | {
"index": 862,
"split": "test"
} |
numina_math | $\frac{2016}{2017^{2}}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_{0}, a_{1}, a_{2}, \\ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \\geqslant 2$ there exists $1 \\leqslant k \\leqslant n$ satisfyin... | math | {
"ground_truth": "$\\frac{2016}{2017^{2}}$",
"style": "rule"
} | {
"index": 863,
"split": "test"
} |
numina_math | $\frac{8}{\sqrt[3]{7}}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the maximal value of\n\n$$\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 nonnegati... | math | {
"ground_truth": "$\\frac{8}{\\sqrt[3]{7}}$",
"style": "rule"
} | {
"index": 864,
"split": "test"
} |
numina_math | 100 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 th... | math | {
"ground_truth": "100",
"style": "rule"
} | {
"index": 865,
"split": "test"
} |
numina_math | $k\left(4 k^{2}+k-1\right) / 2$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | math | {
"ground_truth": "$k\\left(4 k^{2}+k-1\\right) / 2$",
"style": "rule"
} | {
"index": 866,
"split": "test"
} |
numina_math | $t(0,4]$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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$;\... | math | {
"ground_truth": "$t(0,4]$",
"style": "rule"
} | {
"index": 867,
"split": "test"
} |
numina_math | $k=2 n-1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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$ a... | math | {
"ground_truth": "$k=2 n-1$",
"style": "rule"
} | {
"index": 868,
"split": "test"
} |
numina_math | 2013 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 poin... | math | {
"ground_truth": "2013",
"style": "rule"
} | {
"index": 869,
"split": "test"
} |
numina_math | $f(n)=n$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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\nf... | math | {
"ground_truth": "$f(n)=n$",
"style": "rule"
} | {
"index": 870,
"split": "test"
} |
numina_math | 1 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"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 ... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 871,
"split": "test"
} |
numina_math | $f(z)=z$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all functions $f$ from the set of positive integers into the set of positive integers such that for all $x$ and $y$ there exists a non degenerated triangle with sides o... | math | {
"ground_truth": "$f(z)=z$",
"style": "rule"
} | {
"index": 872,
"split": "test"
} |
numina_math | $N(n)=\left\lfloor\frac{2 n}{3}\right\rfloor+1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For any integer $n \\geq 2$, let $N(n)$ be the maximal number of triples $\\left(a_{i}, b_{i}, c_{i}\\right), i=1, \\ldots, N(n)$, consisting of nonnegative integers $a_{i}, b_{i... | math | {
"ground_truth": "$N(n)=\\left\\lfloor\\frac{2 n}{3}\\right\\rfloor+1$",
"style": "rule"
} | {
"index": 873,
"split": "test"
} |
numina_math | 996000 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "On a $999 \\times 999$ board a limp rook can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, an... | math | {
"ground_truth": "996000",
"style": "rule"
} | {
"index": 874,
"split": "test"
} |
numina_math | $90^{\circ}$,$60^{\circ}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A B C$ be a triangle with $A B=A C$. The angle bisectors of $A$ and $B$ meet the sides $B C$ and $A C$ in $D$ and $E$, respectively. Let $K$ be the incenter of triangle $A D... | math | {
"ground_truth": "$90^{\\circ}$,$60^{\\circ}$",
"style": "rule"
} | {
"index": 875,
"split": "test"
} |
numina_math | 1,2,3,4 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all positive integers $n$ such that there exists a sequence of positive integers $a_{1}, a_{2}, \\ldots, a_{n}$ satisfying\n\n$$\na_{k+1}=\\frac{a_{k}^{2}+1}{a_{k-1}+1}-1\n$... | math | {
"ground_truth": "1,2,3,4",
"style": "rule"
} | {
"index": 876,
"split": "test"
} |
numina_math | 6 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they h... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 877,
"split": "test"
} |
numina_math | 180180 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the coordinate plane consider the set $S$ of all points with integer coordinates. For a positive integer $k$, two distinct points $A, B \\in S$ will be called $k$-friends if t... | math | {
"ground_truth": "180180",
"style": "rule"
} | {
"index": 878,
"split": "test"
} |
numina_math | $2^{k-n}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ and $k$ be fixed positive integers of the same parity, $k \\geq n$. We are given $2 n$ lamps numbered 1 through $2 n$; each of them can be on or off. At the beginning all... | math | {
"ground_truth": "$2^{k-n}$",
"style": "rule"
} | {
"index": 879,
"split": "test"
} |
numina_math | $f(x)=x-1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ that satisfy the conditions\n\n$$\nf(1+x y)-f(x+y)=f(x) f(y) \\text { for all } x, y \\in \\mathbb{R}\n$$\n\nand $f(-... | math | {
"ground_truth": "$f(x)=x-1$",
"style": "rule"
} | {
"index": 880,
"split": "test"
} |
numina_math | $\lfloor\frac{2 n-1}{5}\rfloor$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n \\geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\\{1,2, \\ldots, n\\}$ such that the sums of the different pairs are different... | math | {
"ground_truth": "$\\lfloor\\frac{2 n-1}{5}\\rfloor$",
"style": "rule"
} | {
"index": 881,
"split": "test"
} |
numina_math | $\frac{4 \cdot 999^{4}}{27}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In a $999 \\times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $\\left(C_{1}, C_{2}, C_{3}\\right)$ of cells, the first... | math | {
"ground_truth": "$\\frac{4 \\cdot 999^{4}}{27}$",
"style": "rule"
} | {
"index": 882,
"split": "test"
} |
numina_math | 4022 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Players $A$ and $B$ play a game with $N \\geq 2012$ coins and 2012 boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least 1... | math | {
"ground_truth": "4022",
"style": "rule"
} | {
"index": 883,
"split": "test"
} |
numina_math | $(2,251,252)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all triples $(x, y, z)$ of positive integers such that $x \\leq y \\leq z$ and\n\n$$\nx^{3}\\left(y^{3}+z^{3}\\right)=2012(x y z+2) \\text {. }\n$$ Let's think step by step ... | math | {
"ground_truth": "$(2,251,252)$",
"style": "rule"
} | {
"index": 884,
"split": "test"
} |
numina_math | $f(x)=x,f(x)=-x$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all functions $f: \\mathbb{Q} \\rightarrow \\mathbb{Q}$ such that the equation\n\nholds for all rational numbers $x$ and $y$.\n\n$$\nf(x f(x)+y)=f(y)+x^{2}\n$$\n\nHere, $\\m... | math | {
"ground_truth": "$f(x)=x,f(x)=-x$",
"style": "rule"
} | {
"index": 885,
"split": "test"
} |
numina_math | $2021 \cdot 505 \cdot 337$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane, such that\n\n(i) no three points in $P$ lie on a line and\n\n(ii) no two points i... | math | {
"ground_truth": "$2021 \\cdot 505 \\cdot 337$",
"style": "rule"
} | {
"index": 886,
"split": "test"
} |
numina_math | $k=3$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer $k$ for which there exist a colouring of the positive integers $\\mathbb{Z}_{>0}$ with $k$ colours and a function $f: \\mathbb{Z}_{>0} \\righta... | math | {
"ground_truth": "$k=3$",
"style": "rule"
} | {
"index": 887,
"split": "test"
} |
numina_math | $6m$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $m$ be a positive integer. Consider a $4 m \\times 4 m$ array of square unit cells. Two different cells are related to each other if they are in either the same row or in the... | math | {
"ground_truth": "$6m$",
"style": "rule"
} | {
"index": 888,
"split": "test"
} |
numina_math | 1,2 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $m>1$ be an integer. A sequence $a_{1}, a_{2}, a_{3}, \\ldots$ is defined by $a_{1}=a_{2}=1$, $a_{3}=4$, and for all $n \\geq 4$,\n\n$$\na_{n}=m\\left(a_{n-1}+a_{n-2}\\right)... | math | {
"ground_truth": "1,2",
"style": "rule"
} | {
"index": 889,
"split": "test"
} |
numina_math | $2^{n}-n-1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The $n$ contestants of an EGMO are named $C_{1}, \\ldots, C_{n}$. After the competition they queue in front of the restaurant according to the following rules.\n\n- The Jury choo... | math | {
"ground_truth": "$2^{n}-n-1$",
"style": "rule"
} | {
"index": 890,
"split": "test"
} |
numina_math | $(0,1,1),(0,-1,-1),(1,0,1),(-1,0,-1),(1,1,0)$,$(-1,-1,0),\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$,$\left(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all triples $(a, b, c)$ of real numbers such that $a b+b c+$ $c a=1$ and\n\n$$\na^{2} b+c=b^{2} c+a=c^{2} a+b \\text {. }\n$$ Let's think step by step and output the final a... | math | {
"ground_truth": "$(0,1,1),(0,-1,-1),(1,0,1),(-1,0,-1),(1,1,0)$,$(-1,-1,0),\\left(\\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}}\\right)$,$\\left(-\\frac{1}{\\sqrt{3}},-\\frac{1}{\\sqrt{3}},-\\frac{1}{\\sqrt{3}}\\right)$",
"style": "rule"
} | {
"index": 891,
"split": "test"
} |
numina_math | $\frac{n(n+1)}{2}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ be a positive integer. Dominoes are placed on a $2 n \\times 2 n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. Fo... | math | {
"ground_truth": "$\\frac{n(n+1)}{2}$",
"style": "rule"
} | {
"index": 892,
"split": "test"
} |
numina_math | $N=n$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a positive integer $n \\geq 2$, determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_{0}, a_{1}, \\ldots, a_{N}$ such that\n\n(1) $a_{0}+a... | math | {
"ground_truth": "$N=n$",
"style": "rule"
} | {
"index": 893,
"split": "test"
} |
numina_math | 11,13 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all integers $m$ for which the $m \\times m$ square can be dissected into five rectangles, the side lengths of which are the integers $1,2,3, \\ldots, 10$ in some order... | math | {
"ground_truth": "11,13",
"style": "rule"
} | {
"index": 894,
"split": "test"
} |
numina_math | $2^{k-1}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $k$ be a positive integer. Lexi has a dictionary $\\mathcal{D}$ consisting of some $k$-letter strings containing only the letters $A$ and $B$. Lexi would like to write either... | math | {
"ground_truth": "$2^{k-1}$",
"style": "rule"
} | {
"index": 895,
"split": "test"
} |
numina_math | 3 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In an increasing sequence of numbers with an odd number of terms, the difference between any two consecutive terms is a constant $d$, and the middle term is 302 . When the last 4... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 896,
"split": "test"
} |
numina_math | 10,11,12,13,14,-2,-1,0,1,2 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are two increasing sequences of five consecutive integers, each of which have the property that the sum of the squares of the first three integers in the sequence equals th... | math | {
"ground_truth": "10,11,12,13,14,-2,-1,0,1,2",
"style": "rule"
} | {
"index": 897,
"split": "test"
} |
numina_math | 2 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $f(t)=\\sin \\left(\\pi t-\\frac{\\pi}{2}\\right)$, what is the smallest positive value of $t$ at which $f(t)$ attains its minimum value? Let's think step by step and output t... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 898,
"split": "test"
} |
numina_math | -1,0,1 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all integer values of $x$ such that $\\left(x^{2}-3\\right)\\left(x^{2}+5\\right)<0$. Let's think step by step and output the final answer within \\boxed{}.",
"role... | math | {
"ground_truth": "-1,0,1",
"style": "rule"
} | {
"index": 899,
"split": "test"
} |
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