prompt listlengths 1 1 | label stringlengths 1 23 |
|---|---|
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 $z_{1}, z_{2}$ 满足 $\\left|z_{1}\\right|=\\left|z_{1}+z_{2}\\right|=3,\\left|z_{1}-z_{2}\\right|=3 \\sqrt{3}$, 则 $\\log _{3... | 4000 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将正奇数集合 $\\{1,3,5, \\cdots\\}$ 由小到大按第 $n$ 组有 $(2 n-1)$ 个奇数进行分组:\n$\\{1\\}, \\quad\\{3,5,7\\}, \\quad\\{9,11,13,15,17\\}, \\cdot... | 32 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n整数 $\\left[\\frac{10^{93}}{10^{31}+3}\\right]$ 的末两位数是 $\\qquad$.\n\nRemember to put your answer on its own line after \"Answer... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n三位数 $(100,101, \\ldots, 999)$ 共 900 个，在卡片上打印这些三位数，每张卡片上打印一个三位数，有的卡片所印的，倒过来看仍为三位数，如 198 倒过来看是 861 ；有的卡片则不然，如 531 倒过来看是,因此，有些卡片可... | 34 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知点集 $A=\\left\\{(x, y) \\left\\lvert\\,(x-3)^{2}+(y-4)^{2} \\leq\\left(\\frac{5}{2}\\right)^{2}\\right.\\right\\}, B=\\left\\... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将与 105 互素的所有正整数从小到大排成数列, 试求出这个数列的第 1000 项.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "us... | 2186 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在直角坐标平面上, 满足不等式组 $\\left\\{\\begin{array}{l}y \\leqslant 3 x, \\\\ y \\geqslant \\frac{1}{3} x, \\\\ x+y \\leqslant 100\\end{a... | 2551 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求一切实数 $p$ ，使得三次方程 $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ 的三个根均为自然数。\n\nRemember to put your answer on its own line after \"A... | 76 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p(x)=2+4x+3x^2+5x^3+3x^4+4x^5+2x^6$. For $k$ with $0<k<5$, deline\n$$I_k=\\int_{0}^{\\infty}\\frac{x^k}{p(x)}dx$$.\nFor w... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the additive group of ordered pairs of integers ( $m, n$ ) [with addition defined componentwise: $\\left.(m, n)+\\left(m^{\... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCall a set of positive integers \"conspiratorial\" if no three of them are pairwise relatively prime. (A set of integers is \"... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCall a set of positive integers \"conspiratorial\" if no three of them are pairwise relatively prime. (A set of integers is \"... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind positive integers $n$ and $a_1, a_2,…,a_n$ such that\n$$a_1+a_2+…+a_n=1979$$\nand the product $a_1a_2…a_n$ is as large as... | 660 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40}, 20^{30}$?\... | 2301 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo distinct squares of the 8 by 8 chessboard C are said to be adjacent if they have a vertex or side in common. Also, g is ca... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite \"word... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $V$ be the collection of all quadratic polynomials $P$ with real coefficients such that\n$$\\sup\\{\\mid P^{\\prime}(0)\\m... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind\n$$\\lim_{n\\to\\infty}\\left[\\frac{1}{n^{5}}\\sum_{h=1}^{n}\\sum_{k=1}^{n}(5h^{4}-18h^{2}k^{2}+5k^{4})\\right].$$\n\nRe... | -1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind\n$$\n\\lim _{n \\rightarrow \\infty}\\left[\\frac{1}{n^{5}} \\sum_{h=1}^{n} \\sum_{k=1}^{n}\\left(5 h^{4}-18 h^{2} k^{2}+... | -1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n某班有 47 个学生，所用教室有 6 排，每排有 8个座位，用 $(i, j)$ 表示位于第 $i$ 排第 $j$ 列的座位。新学期准备调整座位，设某学生原来的座位为 $(i, j)$ ，如果调整后的座位为 $(m, n)$ ，则称该生作了移动 $[a... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最小的正整数 $m$, 使得对于任意大于 3 的质数 $p$ ，都有 $1051\\left(9^{p^{2}}-29^{p}+m\\right)$ 。\n\nRemember to put your answer on its own line a... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n双曲线 $x^{2}-y^{2}=1$ 的右半支与直线 $x=100$ 围成的区域内部（不含边界）整点（纵横坐标均为整数的点）的个数是\n\nRemember to put your answer on its own line after \"Ans... | 9800 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对于集合 $\\{x \\mid a \\leqslant x \\leqslant b\\}$ ，我们把 $b-a$ 称为它的长度.设集合 $A=\\{x \\mid a \\leqslant x \\leqslant$ $a+1981\\}, B=... | 983 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $100 \\times 25$ 的长方形表格中每一格填入一个非负实数，第 $i$ 行第 $j$ 列中填入的数为 $x_{i, j}$ （ $i=1$ ， $2 ， \\ldots ， 100 ; j=1,2 ， \\ldots, 25 ） （$ ... | 97 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 $z=\\cos \\theta+\\mathrm{i} \\sin \\theta\\left(0^{\\circ} \\leqslant \\theta \\leqslant 180^{\\circ}\\right)$, 复数 $z 、(1... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n如果: \n(1) $a 、 b 、 c 、 d$ 都属于 $\\{1,2,3,4\\}$; \n(2) $a \\neq b, b \\neq c, c \\neq d, d \\neq a$; \n(3) $a$ 是 $a 、 b 、 c 、... | 28 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数\n\n$f(x)= \\begin{cases}\\frac{1}{p}, & x=\\frac{q}{p} ; \\\\ 0, & x \\neq \\frac{q}{p},\\end{cases}$\n\n其中, $p 、 q$ 互质, 且... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知三个互不相等的整数 $x 、 y 、 z$ 之和介于 40 与 44 之间. 若 $x, y, z$ 依次构成公差为 $d$ 的等差数列， $x+y 、 y+z 、 z+x$ 依次构成公比为 $q$ 的等比数列, 则 $d q=$.\n\nReme... | 42 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正六边形 $A B C D E F$ 的边长为 1. 则 $(\\overrightarrow{A B}+\\overrightarrow{D C}) \\cdot(\\overrightarrow{A D}+\\overrightarrow{B E... | -3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系中, $A(1,2), B(3,0)$, 点 $P$ 为圆 $(x-3)^{2}+(y-2)^{2}=1$ 上任意一点，设 $\\overrightarrow{O P}=\\lambda \\overrightarrow{O A}+\\... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数\n\n$$f(\\:x\\:)=\\sqrt{2x^2\\:+2x\\:+41}\\:-\\sqrt{2x^2\\:+4x\\:+4}\\:(\\:x\\:\\in\\:\\mathbf{R}\\:)\\:.$$\n则$f(x)$的最大值为\n... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $3\\times3$ 方格表中 ， 若九个格中的 每一个均被染成红 、 黄、 蓝 、 紫 四种颜色之一 ,任意 两个有公共边 的 格 的 染色均不相同 ,且该方格表 中 恰有两个格染蓝色 , 则符合条件的染色方法种数为$\\qquad$ (用具体... | 3816 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 $f:\\{1,2, \\cdots, 10\\} \\rightarrow\\{1,2, \\cdots, 5\\}$ ，且对一切 $k=1,2, \\cdots, 9$ ，有\n$|f(k+1)-f(k)| \\geqslant 3$.\... | 288 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $(-x y+2 x+3 y-6)^{6}$ 的展开式中, $x^{4} y^{3}$ 的系数为 $\\qquad$. (用具体数字作答)\n\n\nRemember to put your answer on its own line after... | -21600 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $x$ 为实数, 定义 $\\lceil x\\rceil$ 为不小于实数 $x$ 的最小整数 (如 $\\rceil\\rceil=4,\\lceil-\\pi\\rceil=-3$ ). 则关于实数 $x$ 的方程\n$\\lceil 3 x+... | -4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$\\left(x^{2}+x-\\frac{1}{x}\\right)^{6}$ 的展开式中的常数项是 $\\qquad$ (用具体数字作答). \n\nRemember to put your answer on its own line aft... | -5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设等比数列$\\left\\{a_n\\right\\}$的前$n$项和为$S_n,$满足 $S_{n}=\\frac{(a_{n}+1)^{2}}{4}.$则 $S_{20}$的值为$\\qquad$.\n\nRemember to put your... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若正整数 $m 、 n$ 满足 $\\frac{(m+n)!}{n!}=5040$,则 $m!n$ 的值为 $\\qquad$。\n\nRemember to put your answer on its own line after \"Answer... | 144 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many positive integers $N$ satisfy all of the following three conditions?\n(i) $N$ is divisible by 2020 .\n(ii) $N$ has at... | 508536 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length 5, and after each... | 578 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$\\operatorname{Let}\\left(x_{1},y_{1}\\right)=\\left(0.8,0.6\\right)$and let $x_{n+1}=x_{n}\\cos y_{n}-y_{n}\\sin y_{n}$ and ... | -1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p(x)$ be a nonzero polynomial of degree less than 1992 having no noncon-\nstant factor in common with $x^3-x.$ Let\n\n$$\... | 3984 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13)=1+1+0... | 1990 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of 2023·n. What is the minimum va... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $P_n$ be the number of permutations $\\pi$ of $\\{1,2,\\ldots,n\\}$\nsuch that\n$|i-j|=1$ implies $|\\pi(i)-\\pi(j)|\\le... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDefine $C(\\alpha)$ to be the coefficient of $x^{1992}$ in the power series expansion about $x=0$ of $(1+x)^{\\alpha}$. Evalua... | 1992 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n我们称 $A_{1}, A_{2}, \\cdots, A_{n}$ 为集合 $A$ 的一个 $n$ 分划，如果 \n(1) $A_{1} \\cup A_{2} \\cup \\cdots \\cup A_{n}=A$; \n(2) $A_{i}... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一个$100\\times100$方格表的左上角小方格中有一只老鼠，右下角小方格中有一块奶酪。老鼠希望移动到右下角小方格中吃奶酪，每次可以从一个小方格移动到相邻的小方格（两个小方格相邻指它们有公共边）。现在在一些小方格的边上放置隔板，老鼠在移动时不能越... | 2220 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n某个国家有2023个岛和2022座桥，任意一座桥连接两个不同的岛，任意两个岛之间至多有一座桥相连，且可以从任意一个岛通过若干座桥到达其他任何岛。若三个岛中的某个岛与另两个岛都有桥连接，则称这三个岛组成“岛群”.已知任两个“岛群”中都有相同的岛，那么恰有... | 1011 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将 $1,2,3,4,5,6,7,8$ 分别放在正方体的八个顶点上，使得每一个面上的任意三个数之和均不小于 10 。求每一个面上四个数之和的最小值。\n\nRemember to put your answer on its own line afte... | 16 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求所有的整数 $n$ ，使得 $n^{4}+6 n^{3}+11 n^{2}+3 n+31$ 是完全平方数.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有 $n(n>12)$ 个人参加某次数学邀请赛，试卷由十五道填空题组成，每答对一题得 1 分，不答或答错得 0 分。分析每一种可能的得分情况发现：只要其中任意 12 个人得分之和不少于 36 分，则这 $n$ 个人中至少有 3 个人答对了至少三道同样的... | 911 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n集合 $I=\\{1,2, \\cdots, 2020\\}$. 称\n$$\n\\begin{aligned}\n& W=\\{w(a, b)=(a+b)+a b \\mid a, b \\in I\\} \\cap I, \\\\\n& Y=\\{... | 2020 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n今有 $n$ 个人排成一个横队, 从左到右按 $1,2, \\cdots, n$ 报数，凡报奇数者退出队伍；剩下的人再从右到左按 $1,2, \\cdots$ 重新报数，凡报奇数者退出队伍；然后剩下的人又从左到右重新报数。如此进行，使得最后队伍中只剩下... | 1016 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n小陶同学玩如下游戏：取定大于 1 的常数 $v$ ；对正整数 $m$ ，游戏的第 $m$ 轮与第 $m+1$ 轮间隔为 $2^{-m}$ 单位时长；第 $m$ 轮在平面上有策略地取一个半径为 $2^{-m+1}$ 的圆形安全区域（含边界，取圆时间忽略不... | 18 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $1,2, \\cdots, 9$ 的所有排列 $X=(x_{1}, x_{2}, \\cdots, x_{9})$ 的集合为 $A$ 。对任意的 $X \\in A$ ，记\n$$\n\\begin{aligned}\n& f(X)=x_{1}+... | 121 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求正整数 $n$ 的最小值, 使得\n$$\n\\begin{aligned}\n& \\sqrt{\\frac{n-2011}{2012}}-\\sqrt{\\frac{n-2012}{2011}} \\\\\n& <\\sqrt[3]{\\frac... | 4024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知奇素数 $x 、 y 、 z$ 满足\n$$\nx\\left|\\left(y^{5}+1\\right), y\\right|\\left(z^{5}+1\\right), z \\mid\\left(x^{5}+1\\right)\n$$\n... | 2013 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n青青草原上生活着编号为 $1,2, \\cdots, 7$的 $7$ 只羊和编号为 $1,2, \\cdots, 2017$ 的 $2017$ 匹狼。在该草原上有如下奇怪的规则： \n（1）定义 $P(n)$ 为小于 $n$ 的素数个数，仅当 $P(... | 2016 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最大的实数$C$，使得不等式\n$$\\sum_{k=1}^{100}\\frac{1}{x_{k}}\\cdot\\sum_{k=1}^{100}\\left(x_{k}^{3}+2x_{k}\\right)-\\left(\\sum_{k=1}^... | 33340000 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $n$ 次实系数多项式 $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\\cdots+a_{1} x+a_{0}$ 满足:对任意 $x \\in[0,1] \\cup[99,100]$ ，都有 $-1 \\leq f(x) ... | 49 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若集合 $X \\subseteq\\{1,2, \\cdots, n\\}$ 满足: 在 $X$ 的任意五元子集 $A$ 中都可以找到两个元素 $a 、 b(a<b)$ 使得 $a \\mid b$, 则记 $|X|$ 的最大可能值是 $f(n)$ ... | 128 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n某国足协规定：在联赛中比赛双方胜一场得$a$分，平一场得$b$分，负一场得0分，其中，实数$a>b>0$。若一支球队经过$n$场比赛后总得分恰有2015种可能，求$n$的最小值。\n\nRemember to put your answer on it... | 62 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知数列 $\\left\\{x_{n}\\right\\}$ 满足:\n$x_{1}=a, x_{2}=b, x_{n+2}=3 x_{n+1}+2 x_{n}$.\n若存在整数 $k \\geqslant 3$, 使得 $x_{k}=2019$, ... | 370 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n长春嘉年华 $G$ 共有 101 个景点，任意两个景点之间总可以通过一些栈道相连，且每条栈道恰连接两个景点，每两个景点之间至多有一条栈道. 若从某个景点出发，沿一系列不同栈道经过不同景点之后回到了该景点, 则把这样的一系列栈道称为 \"环线\", 并将... | 121 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知集合 $A=\\{1,2, \\cdots, 10\\}$. 定义函数 $f: A \\rightarrow A$ 满足: \n(1) 对于任意的 $x 、 y \\in A$, 若 $x \\leqslant y$, 则 \n$f(x) \\... | 17160 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n定义运算 $a * b=a b-5\\left[\\frac{a b}{5}\\right]$ ，其中， [ $x$ ] 表示不超过实数 $x$ 的最大整数, 集合 $A=$ $\\{0,1,2,3,4\\}$ ，双射 $f: A \\rightarr... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设\n$$\n\\begin{aligned}\nf(x)= & a_{2020} x^{2020}+a_{2019} x^{2019}+\\cdots+ \\\\\n& a_{2018} x^{2018}+a_{1} x+a_{0},\n\\end{... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设方程 $x y z=900$ 的全部正整数解为 $\\left(x_{i}, y_{i}, z_{i}\\right)(1 \\leqslant i \\leqslant n)$. 则 $\\sum_{k=1}^{n}\\left(x_{k c}+y... | 22572 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有 2023 个顶点的完全图\n(即任意两个点之间均有连线), 现将每条边进行染色，有公共点的任意两条边颜色不同，定义每个点所连出的边的颜色为一个颜色集。求不同颜色集个数的最小值。\n\nRemember to put your answer on i... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n无锡市有100个景点，各景点每接待1名游客则景点在旅游局积1分。现在共有100名游客在100天的旅游季内参观这些景点。每天若干人出门参观互不相同的景点，且每名游客自行决定参观一个景点或休息不参观。在旅游季结束时，旅游局发现任何一名游客未曾在不同的两天里... | 1000 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设$\\boldsymbol{u}=\\left(u_{1},u_{2},u_{3}\\right)$和$\\boldsymbol{v}=$$\\left(v_{1},v_{2},v_{3}\\right)$是空间向量，满足$u_{i}、v_{i}(i... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n小明在两个 $3 \\times 3$ 方格表 $A 、 B$ 中随机填入 $1,2, \\cdots, 9$ ，使得每个数字恰在每个方格表中出现一次。对于 $1 \\leqslant i<j \\leqslant 9$ ，若 $i 、 j$ 在 $A... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n全集 $U=\\{1,2, \\cdots, 50\\}, A=\\{a, b\\}$ 。若集合 $C_{U} A$ 中所有元素之和等于 $a b$, 则 $a b=$\n$\\qquad$.\n\nRemember to put your answe... | 1204 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将两个相同的白球、三个相同的红球、四个相同的黑球全部放入三个不同的袋子 \n中. 则没有空袋的放法数为 $\\qquad$.\n\nRemember to put your answer on its own line after \"Answer:... | 723 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $S_{n}$ 为数列 $\\left\\{a_{n}\\right\\}$ 的前 $n$ 项和，规定 $S_{0}=0$ 。若对任意的 $n \\in \\mathbf{Z}_{+}$，均有 \n$$ \n\\begin{aligned} ... | -4034 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$\\cos18^{\\circ}+2\\cos36^{\\circ}+\\cdots+20\\cos360^{\\circ}$\n=_______。\n\nRemember to put your answer on its own line aft... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知点$P(4,2)$，过点$P$的直线$l$与$x$轴、$y$轴的正半轴分别交于$A、B$两点，$O$为坐标原点。则$\\triangle AOB$周长的最小值为_______。\n\nRemember to put your answer on i... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设$a_{1},a_{2},\\cdots,a_{k}\\left(k\\in\\mathbf{Z}_{+}\\right)$均为大于1的整数，且满足$$\\left(a_{1}!\\right)\\left(a_{2}!\\right)\\cdots... | 5024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知$f:\\mathbf{Z}\\rightarrow\\mathbf{Z}$为偶函数，且$f(1)=1,f(2017)\\neq1$。若对任意的$x、y\\in\\mathbf{Z}$均满足\n$$2f(x+y)-f(x)-f(y)\\leqsla... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知数列$\\left\\{a_{n}\\right\\}$的前$n$项和为$S_{n}$，且\n$a_{1}=3,S_{n}=2a_{n}+\\frac{3}{2}\\left((-1)^{n}-1\\right)$。\n若$\\left\\{a_{... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知不等式 $\\left|a x^{2}+b x+a\\right| \\leqslant x$ 对 $x \\in$ $[1,2]$ 恒成立. 则 $3 a+b$ 的最大值为 _____.\n\nRemember to put your answe... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $a_{n}$ 是与 $\\sqrt{\\frac{n}{2}}$ 的差最小的整数， $b_{n}$ 是与 $\\sqrt{2 n}$ 的差最小的整数，记 $\\left\\{\\frac{1}{a_{n}}\\right\\}$ 的前 $n$ 项... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $f(x)=x^{2}-53 x+196+\\left|x^{2}-53 x+196\\right|$, 求 $f(1)+f(2)+\\cdots+$ $f(50)$ 的值。\n\nRemember to put your answer on i... | 660 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求 $k$ 的最小值，使得 $7 \\times 7$ 方格表中挖去 $k$ 个格后，不存在 $T$ 字型 $(T$ 字型指一个方格及三个与其有公共边的方格构成的图形)。\n\nRemember to put your answer on its ow... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind positive integers $n$ and $a_{1}, a_{2}, \\ldots, a_{n}$ such that\n$$\na_{1}+a_{2}+\\cdots+a_{n}=1979\n$$\nand the produ... | 660 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 $f: \\mathbf{R} \\rightarrow \\mathbf{R}$, 若对任意实数 $x 、 y 、 z$都有 \n \n$$ \n\\begin{aligned} \n& \\frac{1}{3} f(x y)+\\... | 1650 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $f(x)$ 是二次函数, 且 $2 x^{2}-4 x+3 \\leqslant f(x) \\leqslant 3 x^{2}-6 x+4, f(3)=11$, 则 $f(5)=$?.\n\nRemember to put your answe... | 41 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若实数 $x, y$ 满足 $4 x^{2}-4 x y+2 y^{2}=1$ ，则 $3 x^{2}+x y+y^{2}$ 的最大值与最小值的和为？\n\nRemember to put your answer on its own line aft... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知抛物线 $\\Omega: y=x^{2}$, 动线段 A B 在直线 $y=\\sqrt{3} x-3$ 上 ( B 在 A 右侧), 且 $|A B|=2 \\sqrt{3}$. 过 A 作 $\\Omega$的切线, 取左边的切点为 M. 过... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAlice drew a regular 2021-gon in the plane. Bob then labelled each vertex of the 2021-gon with a real number, in such a way th... | 2018 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvery positive integer is marked with a number from the set $\\{0,1,2\\}$, according to the following rule: \nif a positive i... | 2021 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $a_{1} 、 a_{2} 、 a_{3} 、 a_{4}$ 均为实数。若集合 $\\left\\{a_{1}, a_{2}, a_{3}, a_{4}\\right\\}$ 的所有非空真子集的元素之和为 28, 则 $a_{1}+a_{2}+... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$x \\&(y \\& z)=x \\& y+z, x \\& x=0$, 求 $2000 \\& 2022$\n\nRemember to put your answer on its own line after \"Answer:\".",
... | -22 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若三个正整数$a,b,c$使得$\\sqrt{a+b}$，$\\sqrt{b+c}$，$\\sqrt{c+a}$是三个连续正整数，则$a^{2}+b^{2}+c^{2}$的最小值是 ？\n\nRemember to put your answer on... | 1297 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若定义在 $\\mathbb{R}$ 上的光滑函数 $h(x)$ 在 $x=0$ 处的泰勒展开为 $c_{k} x^{k}+c_{k+1} x^{k+1}+\\cdots$,其中 $c_{k} \\neq 0$ ，则定义 $\\operatorname... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对非负实数 $x$ 令 $\\pi(x)$ 为不超过 $x$ 的素数个数，如 $\\pi(1)=0, \\pi(2)=1$ ，再令 $f(x)$为第 $\\left[\\frac{x}{4}\\right]+1$ 个素数, 求 $\\int_{0}^{... | 2500 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$I=\\int_{0}^{\\frac{\\pi}{2}} \\frac{1}{1+(\\cos t)^{2}} d t$, 则 $[10 I]=$？\n\nRemember to put your answer on its own line af... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n考虑如下 $2021 \\times 2021$ 的方阵 $M=\\left(\\begin{array}{cccccc}0 & 2 & 3 & 4 & \\cdots & 2021 \\\\ 1 & 0 & 3 & 4 & \\cdots & 202... | 4082420 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.