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 $Answer is the answer to the problem.\n\nFind the greatest five-digit number A in which the fourth digit is greater than the fifth; the third greater than the sum of t... | 95210 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe streets of a city are represented on a map as straight lines that divide a square into 25 smaller squares of side 1. (The ... | 60 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTo guard a military installation around the clock, a day shift and a night shift are required. A sentry guard may take either ... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x_1 = 4, x_2 = 6$ and define $x_n$ for $n ≥ 3$ to be the least non-prime greater than $2x_{n−1} − x_{n−2}.$ Find $x_{1000... | 501500 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n试求满足下列条件的大于 $5$ 的最小奇数 $a$: 存在正整数 $m_{1} 、 n_{1} 、 m_{2} 、 n_{2}$ ，使得\n$a=m_{1}^{2}+n_{1}^{2}, a^{2}=m_{2}^{2}+n_{2}^{2},$\n 且 ... | 261 |
[
{
"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已知平面上有 $100$ 条直线，用 $T$ 表示由这些直线中的某三条直线围成的直角三角形的集合。求 $|T|$ 的最大值。\n\nRemember to put your answer on its own line after \"Answer:\... | 62500 |
[
{
"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一副三色纸牌，共有 32 张，其中红黄蓝每种颜色的牌各 10 张，编号分别是 $1,2, \\cdots, 10$ ；男有大小王牌各一张，编号均为 0 。从这副牌中任取若干张牌，然后按如下规则计算分值：每张编号为 $k$ 的牌计为 $2^{k}$ 分。... | 1006009 |
[
{
"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将一个 $3 \\times 3$ 的正方形的四个角上各去掉一个单位正方形所得到的图形称为\"十字形\"。在一个 $10 \\times 11$ 的棋盘上，最多可以放置多少个互不重叠的 \"十字形\"(每个\"十字形\"恰好盖住棋盘上的 5 个小方格）... | 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\n求同时满足下列两个条件的多项式 $f(x)=a x^{3}+b x$ 的个数:\n(1) $a, b \\in\\{1,2, \\cdots, 2013\\}$;\n(2) $f(1), f(2), \\cdots, f(2013)$ 中任意两数之差不... | 7200 |
[
{
"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求最大正整数 $m$, 使得可以在 $m$ 行 8列的方格表的每个方格中填人 C、G、M、O 这四个字母之一，并且具有如下性质：对于方格表的任意不同两行，至多存在一列，使得这两行在此列处的字母相同。\n\n\nRemember to put your ... | 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考虑所有满足以下两个条件的实数序\n列 $x_{0}, x_{1}, \\cdots, x_{100}$ :\n(1) $x_{0}=0$;\n（2）对任意整数 $i(1 \\leqslant i \\leqslant 100)$ 有\n$1 \\le... | 67 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将 2018 行 4 列方格表的每个格染成红色或蓝色，使得方格表每行的红格数与蓝格数相等，每列的红格数与蓝格数也相等。记 $m$ 为满足上述要求的染色方法的总数。求 $m$除以 2018 所得的余数。\n\nRemember to put your a... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n某个国家有 2023 个岛和 2022 座桥, 任意一座桥连接两个不同的岛, 任意两个岛之间至多有一座桥相连, 且可以从任意一个岛通过若干座桥到达其他任何岛. 若三个岛中的某个岛与另两个岛都有桥连接, 则称这三个岛组成\"岛群\", 已知任两个\"岛群... | 1011 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设集合 $M \\subseteq\\{1,2, \\cdots, 2011\\}$ 满足：在 $M$ 的任意三个元素中都可以找到两个元素 $a 、 b$ ，使得 $a \\mid b$ 或 $b \\mid a$ 。 求 $|M|$ 的最大值（ $|... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求满足下述条件的最小正实数 $k$ ：对任意不小于 $k$ 的 4 个互不相同的实数 $a 、 b$ 、 $c 、 d$, 都存在 $a 、 b 、 c 、 d$ 的一个排列 $p 、 q 、 r 、 s$,使得方程 $\\left(x^{2}+p x... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n我们称 $A_{1}, A_{2}, \\cdots, A_{n}$ 为集合 $A$ 的一个 $n$ 分划, 如果(1) $A_{1} \\cup A_{2} \\cup \\cdots \\cup A_{n}=A$;(2) $A_{i} \\cap ... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $f(x)=\\left[\\frac{x}{1!}\\right]+\\left[\\frac{x}{2!}\\right]+\\cdots+\\left[\\frac{x}{2013!}\\right]$,其中， $[x]$ 表示不超过实数 $... | 587 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设集合 $M=\\{1,2, \\cdots, 10\\}, T$ 为 $M$ 的某些二元子集构成的集合，满足对 $T$ 中任意两个不同的元素 $\\{a, b\\} 、\\{x, y\\}$ ，均有 \n$11 \\nmid(a x+b y)(a ... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$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... | 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一个 $100 \\times 100$ 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... | 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 $Answer is the answer to the problem.\n\n\n$(1)$求不定方程\n$m n+n r+m r=2(m+n+r)$\n的正整数解 $(m, n, r)$ 的组数;\n\n\nRemember to put your answer on its own line after \"Answer:\... | 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在一个圆周上给定 $8$ 个点 $A_{1}, A_{2}, \\cdots$, $A_{8}$. 求最小的正整数 $n$, 使得以这 $8$ 个点为顶点的任意 $n$ 个三角形中，必存在两个有公共边的三角形。 \n\nRemember to put... | 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$2021$ 个单位正方形排成一行，最左侧和最右侧的格均为黑色，其余 $2019$ 个格均为白色。随机选取两个相邻白格，并随机将其中一个白格染黑，重复此过程，直至无两个相邻白格为止。令 $E$ 是剩余白格数目的数学期望。求不超过 $E$ 的最大整数 $... | 743 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将正九边形的每条边和每条对角线都染成红色或蓝色之一，使得每个顶点引出的 $8$ 条线段中恰有 $4$ 条红色线段. \n求三边全是红色的三角形个数的最大值. \n\nRemember to put your answer on its own li... | 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 $p$ 为质数, $\\sqrt{p}$ 的小数部分为 $x, \\frac{1}{x}$ 的小数部分为 $\\frac{\\sqrt{p}-31}{75}$. 求所有满足条件的质数 $p$ 的值.\n\nRemember to put your... | 2011 |
[
{
"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已知正数 $a 、 b 、 c$ 满足 $a+b+c=13$. 则 $\\frac{a^{2}+b^{3}+c^{4}+2019}{10 b+123 c+26}$ 的最小值为 $\\qquad$.\n\nRemember to put your ans... | 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已知抛物线 $\\Gamma: y^{2}=8 x$ 的焦点为 $F$ ，过 $F$ 作一条直线 $l$ 与抛物线 $\\Gamma$ 交于 $A 、 B$ 两点，分别过 $A 、 B$ 作抛物线 $\\Gamma$ 的切线，与 $y$ 轴交于 $P ... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $P(z)=z^{4}-(6 \\mathrm{i}+6) z^{3}+24 \\mathrm{i}z^{2}- (18 \\mathrm{i}-18) z-13$. \n求 $P(z)=0$ 的四个根在复平面上对应的四个点所围成凸四边形的面积... | 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已知整数 $a, b, c, d$ 满足 $a+b+c+d=6$ ，则 $a b+a c+a d+b c+b d+c d$ 的正整数取值个数为 $\\qquad$. \n\nRemember to put your answer on its own... | 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若正方体 $A_{1} A_{2} A_{3} A_{4}-B_{1} B_{2} B_{3} B_{4}$ 的棱长为 1 ，则集合\n$\\left\\{x \\mid x=\\overrightarrow{A_{1} B_{1}} \\cdot \... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设一个袋子里有红、黄、蓝色小球各一个. 现每次从袋子里取出一个球（取出某色球的概率均相同), 确定颜色后放回, 直到连续两次均取出红球时为止, 记此时取出球的次数为 $\\xi$. 则 $\\xi$ 的数学期望为 $\\qquad$.\n\n\nRem... | 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在一个 $3 \\times 3$ 的方格表中, 若 9 个方格中每一个均被染成红、黄、蓝、紫四种颜色之一，任意两个有公共边的小方格的颜色均不相同，且该方格表中恰有 2 个小方格染蓝色，则符合条件的染色方法总数为 $\\qquad$. (用具体数字作答... | 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在 $3 \\times 3$ 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 $\\qquad$ （用具体数字作答）。\n\nRemembe... | 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 $f(x)$ 在区间 $(0,+\\infty)$ 上严 格单调递减，对任意的 $x \\in(0,+\\infty)$ ，均有 \n$$ \nf(x) f\\left(f(x)+\\frac{2}{x}\\right)=\\frac{1... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 $f:\\{1,2, \\cdots, 10\\} \\rightarrow\\{1,2, \\cdots, 5\\}$ ，且对一切 $k=1,2, \\cdots, 9$ ，有 \n$|f(k+1)-f(k)| \\geqslant 3$... | 288 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知数列 $\\left\\{a_{n}\\right\\}$ 满足: \n$$ \n\\begin{aligned} \n& a_{1}=1, a_{2}=2, a_{2 k+1}=\\frac{a_{2 k}^{2}}{a_{2 k-1}} ... | 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\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 $Answer is the answer to the problem.\n\n$R(n)$ 表示正整数 $n$ 除以 $2,3,4,5,6,7,8,9,10$ 的余数之和, 则满足 $R(n)=R(n+1)$ 的两位数 $n$ 的个数为 $\\qquad$. \n\nRemember to put your answer on... | 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集合 $U=\\{1,2, \\cdots, 366\\}$ ，则 $U$ 的互不相交各元素之和为 $17$ 的倍数的二元子集最多有 $\\qquad$个. \n\nRemember to put your answer on its own lin... | 179 |
[
{
"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 $x, y, z$ 均为正整数, 且 $\\frac{x(y+1)}{x-1}, \\frac{y(z+1)}{y-1}, \\frac{z(x+1)}{z-1}$ 均为正整数, 则 $x y z$ 的最大值和最小值之和为 $\\qquad$.\... | 701 |
[
{
"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 $\\triangle A B C$ 中, 若 $O$ 为形外一点, 满足 $\\angle B O C=2 \\angle B A C$, 线段 $O C$ 与线段 $A B$ 交于 $D$, 且 $O B=O C=3$, $O D=2$, 求 ... | 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 $a_{k}$ 为 $\\log _{2} k, \\log _{3} k, \\cdots, \\log _{2018} k$ 中整数项的个数. 则 $\\sum\\limits_{k=1}^{2018} a_{k}=$ \n\nRememb... | 4102 |
[
{
"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 $\\triangle A B C$ 中, 已知 $\\sin A=10 \\sin B \\sin C, \\cos A=10 \\cos B \\cos C$, 则 $\\tan A$ 的值为 $\\qquad$. \n\nRemember t... | 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若平面上有 100 条二次曲线, 则这些曲线可以把平面分成若干个连通区域, 则连通区域数量最大值为 $\\qquad$.\n\nRemember to put your answer on its own line after \"Answer:\".... | 20101 |
[
{
"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 $y_{n}=1 \\overbrace{22 \\cdots 2}^{n} 1$. 若 $10^{9}-1 \\mid y_{n}$, 则 $n$ 的最小值为 $\\qquad$.\n\nRemember to put your answer o... | 80 |
[
{
"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如果一个十位数 $F$ 的各位数字之和为 81 ，则称 $F$ 是一个 \"小猿数\". 则小猿数的个数为 $\\qquad$.\n\nRemember to put your answer on its own line after \"Answer... | 48619 |
[
{
"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 $\\sqrt{\\frac{n}{2}}$ 的差的绝对值最小的整数, $b_{n}$ 是与 $\\sqrt{2 n}$ 的差的绝对值最小的整数. 记 $\\left\\{\\frac{1}{a_{n}}\\right\\}$... | 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设正整数 $m, n \\leqslant 2021$ ，并且 $\\frac{m}{n+1}<\\sqrt{2}<$ $\\frac{m+1}{n}$, 则这样的数组 $(m, n)$ 的个数是 $\\qquad$. \n\nRemember to... | 3449 |
[
{
"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已知不全相等的三个实数 $a, b, c$, 满足 $a=a b+c, b=b c+a, c=c a+b$ ，则 $a+b+$ $c=$ $\\qquad$.\n\nRemember to put your answer on its own line... | 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正整数 $Y=\\sum_{i=0}^{2021}\\left[\\frac{2^{i}}{7}\\right]$ 的个位数字是 $\\qquad$.\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已知数列 $\\left\\{a_{k}\\right\\}_{1 \\leq k \\leq 5}$ 各项均为正整数, 且 $\\left|a_{k+1}-a_{k}\\right| \\leq 1,\\left\\{a_{k}\\right\\}$... | 211 |
[
{
"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 $f(x)$ 是二次函数, $f(-2)=0$, 且 $2 x \\leq f(x) \\leq \\frac{x^{2}+4}{2}$, 则 $f(10)=$ $\\qquad$. \n\nRemember to put your answe... | 36 |
[
{
"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已知正整数 $y$ 不超过 $2022$ 且满足 $100$ 整除 $2^{y}+y$ ，则这样的 $y$ 的个数为 $\\qquad$. \n\nRemember to put your answer on its own line after \... | 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方程 $x+y+z=2010$ 满足 $x \\leqslant y \\leqslant z$ 的正整数解 $(x, y, z)$ 的个数是 $\\qquad$.\n\n\nRemember to put your answer on its own... | 336675 |
[
{
"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 $\\triangle A B C$ 的角 $A, C$ 满足 $5(\\cos A+\\cos C)+4(\\cos A \\cos C+1)=0$, 则 $\\tan \\frac{A}{2} \\cdot \\tan \\frac{C}{2}... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对于任意集合 $S$ ，用 $|S|$ 表示集合 $S$ 中的元素个数，用 $n(S)$ 表示集合 $S$ 的子集个数。若 $A, B, C$ 是三个有限集, 且满足条件:\n\n$(1)$ $|A|=|B|=2016$;\n\n$(2)$ $n(A)... | 2015 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n由 $1,2, \\cdots, 9$ 这九个正整数构成的所有圆排列中, 任意相邻两数之积均不超过 $60$ 的圆排列的个数为 $\\qquad$.\n\nRemember to put your answer on its own line afte... | 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 $Answer is the answer to the problem.\n\n$\\min \\{\\max \\{\\sqrt{x},|x-6|\\}\\}=$ $\\qquad$.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 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已知下列结论成立:\n任意给定复平面 $\\mathbb{C}$ 上的多项式 $f(z)$ 和 $g(z)$ ，以及实数 $r>0$ 。若不等式\n$$\n|g(z)|<|f(z)|\n$$\n对所有满足 $|z|=r$ 的复数 $z$ 都成立，则 $... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求使得不等式\n$$\n2 x^{4}+m x^{3}+(m+6) x^{2}+m x+2>0\n$$\n对所有实数 $x$ 恒成立的正整数 $m$ 的最大值.\n\nRemember to put your answer on its own lin... | 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已知 $A=\\left[\\begin{array}{ccc}2 & 0 & 0 \\\\ -1 & 3 & 1 \\\\ -1 & 1 & 3\\end{array}\\right]$, 且 $A^{10}=\\left[\\begin{array... | 524800 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n记\n$$\nF(x)=16 \\int_{\\frac{\\pi^{2}}{4}+\\cos \\left(3 x+\\frac{\\pi}{2}\\right)}^{\\frac{25 \\pi^{2}}{4}+2 x^{2}} \\sin \\s... | 48 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n记\n$$\na=\\lim _{n \\rightarrow+\\infty} \\sum_{k=0}^{n-1} \\frac{120}{\\sqrt{n^{2}+k n}}\n$$\n求取整 $[a]$ 的值.\n\nRemember to pu... | 90 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n实对称矩阵 $A=\\left[\\begin{array}{cccc}4 & 1 & 1 & 1 \\\\ 1 & 4 & 1 & 1 \\\\ 1 & 1 & 4 & 1 \\\\ 1 & 1 & 1 & 4\\end{array}\\right]... | 67 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $t$ 是一个实数变量. 记矩阵\n$$\n\\left[\\begin{array}{cccc}\nt^{2}+2 t+2 & 0 & 0 & t \\\\\n0 & 2 t+1 & t & t \\\\\n0 & t & 3 t+1 & t \... | 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已知关于实数变量 $x_{1}, x_{2}, x_{3}, x_{4}$ 的线性方程组\n$$\n\\left\\{\\begin{array}{l}\nx_{1}+2 x_{2}+x_{3}+x_{4}=0 \\\\\n3 x_{1}+a x_{2... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $\\forall x \\in R, f(x)=2 x^{4}+m x^{3}+(m+6) x^{2}+m x+2>0$, 求正整数 $m$ 的最大值.\n\nRemember to put your answer on its own lin... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知矩阵\n$$\n\\left[\\begin{array}{cccc}\nt^{2}+2 t+2 & 0 & 0 & t \\\\\n0 & 2 t+1 & t & t \\\\\n0 & t & 3 t+1 & t \\\\\nt & t & t... | 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 $M=\\left(\\begin{array}{cccccc}0 & 2 & 3 & 4 & \\cdots & 2021 \\\\ 1 & 0 & 3 & 4 & \\cdots & 202... | 4082420 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$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 $[10 I]=$ $\\qquad$.\n\nRemember to put your answer on its o... | 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在模 $2520$ 的同余系里，满足方程 $a^{2} \\equiv a(\\bmod 2520)$ 的元素 $a$ 的个数为 $\\qquad$.\n\nRemember to put your answer on its own line aft... | 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已知 $y=f(x)+x^{3}$ 为偶函数, 且 $f(10)=15$, 则 $f(-10)$ 的值为 $\\qquad$.\n\n\nRemember to put your answer on its own line after \"Answe... | 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\n已知 $y=f(x)+x^{3}$ 为偶函数, $f(10)=15$. 则 $f(-10)$ 的值为 $\\qquad$.\n\n\nRemember to put your answer on its own line after \"Answer:... | 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已知数列 $\\left\\{a_{n}\\right\\}$ 是一个等差数列, 首项与公差均为正数, 且 $a_{2}, a_{5}, a_{9}$ 依次成等比数列，则使得 $a_{1}+a_{2}+\\cdots+a_{k}>100 a_{1}$ ... | 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 $f(x), g(x)$ 均为定义在 $R$ 上的函数, $f(x)$ 的图像关于直线 $x=1$ 对称, $g(x)$ 的图像关于点 $(1,-2)$ 中心对称, 且 $f(x)+g(x)=9^{x}+x^{3}+1$, 则 $f(2) g(2... | 2016 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $A$ 是任意一个 11 元实数集合。令集合 $B=\\{u v \\mid u, v \\in A, u \\neq v\\}$ 。求 $B$ 的元素个数的最小值.\n\n\nRemember to put your answer on its ... | 17 |
[
{
"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 整数 $n$ 满足 $n \\leq 2016$, 且 $\\left\\{\\frac{n}{2}\\right\\}+\\left\\{\\frac{n}{4}\\right\\}+\\left\\{\\frac{n}{6}\\right\... | 168 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $A=\\{a \\mid-1 \\leq a \\leq 2\\}$, 则平面点集 $B=\\{(x, y) \\mid x, y \\in A, x+y \\geq 0\\}$ 的面积为 $\\qquad$.\n\n\nRemember to ... | 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设 $A=\\{a \\mid-1 \\leqslant a \\leqslant 2\\}$, 则平面点集 $B=\\{(x, y) \\mid x, y \\in A, x+y \\geqslant 0\\}$ 的面积为 $\\qquad$.\n\... | 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已知复数 $z$ 满足 $z^{2}+2 z=\\bar{z} \\neq z(\\bar{z}$ 表示 $z$ 的共轭复数 $)$ ，则 $z$ 的所有可能值的积为 $\\qquad$.\n\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设整数 $n>4,(x+2 \\sqrt{y}-1)^{n}$ 的展开式中 $x^{n-4}$ 与 $x y$ 两项的系数相等, 则 $n$ 的值为 $\\qquad$.\n\nRemember to put your answer on its ow... | 51 |
[
{
"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设正整数 $a, b, c$ 中任意两个数不互素，且 $a+b+c$ 为素数。求 $a+b+c$ 的最小值。\n\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 31 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$a_{n}$ 是离 $\\sqrt{n}$ 最近的整数, 则 $S_{2021}=$\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "u... | 60585 |
[
{
"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求六个元素的置换群 $S_{6}$ 中 $6$ 阶元素的个数。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] | 240 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$A$ 是 10 阶实矩阵, 且 $A^{5}=O$, 则 $\\operatorname{rank}(A)$ 的所有可能值之和为 $\\qquad$. \n\nRemember to put your answer on its own line ... | 36 |
[
{
"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给定以正方形内的 2022 个点，且这 2022 个点以及正方形的顶点中任三点不共线。在这 2022 以及正方形的顶点之间连一些线段，将正方形划分为一些三角形，并要求这些线段之间只可以在端点处有公共点，则所连线段（不包含原正方形的边）的总数为 $\\q... | 6067 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n记 $\\omega_{n}=\\exp \\left(\\frac{2 \\pi \\sqrt{-1}}{n}\\right)$ 。已知 $n$ 阶傅里叶矩阵定义为 \n$$ \n\\left[\\begin{array}{ccccc} \n1... | 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设函数 $\\varphi_{1}(x), \\varphi_{2}(x)$ 并足 $\\varphi_{i}^{\\prime \\prime}(x)-\\ln (x+1) \\varphi_{i}^{\\prime}(x)+e^{x} \\varp... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正实数 $c_{1}, c_{2}, c_{3}$ 使得对任意定义在 $[-\\pi, \\pi]$ 上的三次多项式 $f(x)$ 均有 $\\frac{1}{2 \\pi} \\int_{-\\pi}^{\\pi} f(x) {d} x=$ $c_... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知欧拉常数 $\\gamma=\\lim _{n \\rightarrow+\\infty}\\left(1+\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{n}-\\ln n\\right)$ 约等于 0.5... | 236 |
[
{
"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已知下列不等式 $\\frac{e^{22 x}+e^{-22 x}}{2} \\leq e^{c x^{2}}$ 对所有实数 $x$ 均成立, 则 $[c]$ 的最小值为 $\\qquad$.\n\nRemember to put your answ... | 242 |
[
{
"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 $\\mathbb{R}^{100}$ 中的点 $x=\\left(x_{1}, x_{2}, \\cdots, x_{100}\\right)$ 和 $y=\\left(y_{1}, y_{2}, \\cdots, y_{100}\\right)... | 198 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$f(x)=\\sum_{n=0}^{\\infty} 2^{-n} \\cos (n x)$, 记 $A=\\int_{0}^{2 \\pi} f^{2}(x) \\mathrm{d} x$, 则 $[10A]$=\n\nRemember to pu... | 73 |
[
{
"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 $\\mathbb{R}$ 上的无穷次可微函数，满足方程 $f^{\\prime \\prime}(x)=x f(x)$ 。已知 $f(0)=18, f^{\\prime}(0)=3$. 设 $f(x)$ 的考克劳林展开式为 $\... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$S=\\int_{-\\infty}^{\\infty} \\frac{1}{1+x^{2}+x^{4}} \\mathrm{~d} x$, 则 $\\left[100 S^{2}\\right]=$ $\\qquad$\n\nRemember to... | 328 |
[
{
"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_{50}^{+\\infty} e^{-x^{2}} \\mathrm{~d} x$, 则 $[-\\ln I]=$ $\\qquad$ \n\nRemember to put your answer on its own line... | 2504 |
[
{
"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我们称置换群 $S_{9}$ 中的两个元素 $\\sigma_{1}, \\sigma_{2}$ 是共轭的，如果存在 $S_{9}$ 中的元素 $\\tau$使得 $\\tau^{-1} \\sigma_{1} \\tau=\\sigma_{2}$ .... | 15120 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.