prompt listlengths 1 1 | label stringlengths 1 23 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n记集合 A=\\{1,2,\\cdots,2023\\}，存在集合 B\\subseteq A，且 B 中任意两个元素之和不是 5 的倍数；存在集合 C\\subseteq B，且 C 中元素均为完全平方数. 则 |C| 的最大值为__________... | 19 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有 8 块不同的积木，每块积木的形状为方形或者圆形，颜色为红色或者黄色，印有城市名为南昌或者北京. 从 8 块不同的积木中按顺序任取 n 块积木（2\\leq n\\leq 8），D_n 表示后一次与前一次取出的积木在形状、颜色、城市名三个方面恰好有两... | 240 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最小的正整数 k，使得对集合 S=\\{1,2,\\cdots,2012\\} 的任意一个 k 元子集 A，都存在 S 的三个不同的元素 a,b,c，使得 a+b, b+c, c+a 均在 A 中.\n\nRemember to put your a... | 1008 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设多项式 f(x)=x^{2020}+\\ds{i=0}{2019}c_ix^i，其中 c_i\\in\\{-1,0,1\\}. 记 N 为 f(x) 的正整数根的个数（含重根）. 若 f(x) 无负整数根，求 N 的最大值.\n\nRemember ... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n某班有 47 个学生，所用教室有 6 排，每排有 8 个座位，用 (i,j) 表示位于第 i 排第 j 列的座位. 新学期准备调整座位，设某学生原来的座位为 (i,j)，如果调整后的座位为 (m,n)，则称该生作了移动 [a,b]=[i-m,j-n]，... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设有理数 r=\\df{p}{q}\\in(0,1)，其中 p,q 为互质的正整数，且 pq 整除 3600. 这样的有理数 r 的个数为__________..\n\nRemember to put your answer on its own li... | 112 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n\\displaystyle\\prod_{k=1}^6(x^5-5x+k)=0 有__________个实根.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定 3\\times3 整数元矩阵M=\\begin{pmatrix} 11&11&20\\\\ 10&10&-8\\\\ -10&2&8\\end{pmatrix},已知存在整数元矩阵 P,Q，满足 \\det P, \\det Q\\in\\{-... | 288 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n令 I=\\displaystyle\\sqrt{\\df{2}{\\pi}}\\int_{-\\infty}^{+\\infty}e^{x-10\\cosh(2x)}\\dd x，则 [|\\ln I|]=__________.\n\nRemembe... | 11 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一次会议共有 24 名代表参加，每两名代表之间要么握手一次，要么不握手. 会议结束后发现，总共出现了 216 次握手，且对任意相互握过手的两名代表 P,Q，在剩下的 22 名代表中，恰与 P,Q 之一握过手的代表不超过 10 名. 一个\"朋友圈\"指... | 864 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最小的正整数 n，使得存在 n 个不同的正整数 s_1, s_2, \\cdots, s_n，满足\\left(1-\\df{1}{s_1}\\right)\\left(1-\\df{1}{s_2}\\right)\\cdots\\left(1-\\... | 39 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对一个立方体，T 为连接两个顶点的线段的集合. 则 T 中垂直线段对的个数减去平行线段对的个数等于__________.\n\nRemember to put your answer on its own line after \"Answer:\".... | 108 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n某马戏团有 n 个小丑，他们每个人均用 12 种不同颜色中的至少 5 种装扮自己. 一天，马戏团经理要求任意两个小丑所用的颜色不能完全相同，并且每种颜色最多只能由 20 个小丑使用. 在满足经理要求的情况下，求 n 的最大值.\n\nRemember ... | 48 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n某次会议共有 30 人参加，已知每个人在其余人中至多有 5 个熟人，且任 5 人中存在两人不是熟人. 求最大的正整数 k，使得在满足上述条件的 30 人中总存在 k 个两两不是熟人.\n\nRemember to put your answer on ... | 6 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n展览馆中有 2015 幅画在展出. 每位观众选择一对画，并将它们的名字写在同一张纸上. 一个艺术家从这张纸中选择一些画对，并评判每对中哪幅画更好. 之后助手按下述规则继续评判：在每一步中，如果对某个正整数 n，在画对 (A_1,A_2) 中 A_1 比... | 2014 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设实数 a,b,c,d 满足 b-d\\geq5，实数 x_1,x_2,x_3,x_4 是多项式 P(x)=x^4+ax^3+bx^2+cx+d 的根，求 (x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1) 的最小值.\n\nRe... | 16 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知一个凸 2010 边形的任意三条对角线不在其内部交于一点. 考虑经过 2010 边形的每个顶点恰一次的闭折线，求该闭折线自交点的数目的最大值.注：闭折线是指由点 P_1,P_2,\\cdots,P_n 及线段 P_1P_2, P_2P_3, \\c... | 2016031 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有 10 个互不相同的正实数，考虑它们两两的和与积（各 45 个）. 已知这些和中有 5 个相等，求最大的正整数 k，使得这些积中有 k 个相等.\n\nRemember to put your answer on its own line after... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设数列 \\{a_n\\} 满足 a_1=8，且对任意正整数 n，均有a_{n+1}=\\left(1+\\df{3}{n}\\right)a_n+\\left(1-\\df{3}{n}\\right)\\cdot 2^n,则 a_{2023} 除以 ... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 f(x)\\in C^{\\infty}(\\bR)，满足 f''(x)=xf(x)，且 f(0)=18，f'(0)=3. 设 f(x) 的泰勒展开式为 \\ds{n=0}{\\infty}a_nx^n，求 \\left[\\df{a_9}{a_{... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 S=\\int_{-\\infty}^{+\\infty}\\df{\\dd x}{1+x^2+x^4}，求 [100S^2].\n\nRemember to put your answer on its own line after \"Answ... | 328 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n令 f(x)=[x]+[2x]+[3x]+[4x]+[5x]，当 x 取遍 [0,50] 中的实数时，求 f(x) 能取到的整数值的个数.\n\nRemember to put your answer on its own line after \"A... | 501 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设S=\\{(x,y,z,w)\\in\\bZ^4\\ |\\ x,y,z,w\\in[0,18],x^2+y^2\\equiv z^2+w^2\\pmod{19}\\},求 |S|.\n\nRemember to put your answer on... | 7201 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 V 为所有次数不超过 100 的实系数多项式构成的线性空间，令V_1=\\{f\\in V\\ |\\ f^{(2k-1)}(0)=0,k=1,2,\\cdots,50\\},V_2=\\{f\\in V\\ |\\ f^{(k)}(-1)=f^{... | 86 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对空间直角坐标系 xyz，先以 z 轴为中心轴旋转 \\df{\\pi}{3}，再以 x 轴为中心轴旋转 \\df{\\pi}{3}，得到一个以某条直线为中心轴的旋转. 设新的旋转角为 \\theta\\in(0,\\pi]. 求 [\\tan^2\\... | 63 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对 \\bR^{100} 中的点 x=(x_1,x_2,\\cdots,x_{100}), y=(y_1,y_2,\\cdots,y_{100})，令 f(x,y)=\\sqrt{\\ds{i=1}{100}(x_i-y_i)^2}. 对 x\\neq... | 198 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数 f(x)=\\ln|x|+|x|-2 的零点都在区间 [a,b] (a,b\\in\\bZ, a<b) 内，则 b-a 的最小值为__________.\n\nRemember to put your answer on its own lin... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n用 f(X,\\Gamma) 表示点 X 与曲线 \\Gamma 上任意一点距离的最小值. 已知 \\odot O:x^2+y^2=1 及 \\odot O_1:(x-4)^2+y^2=4. 设 P 为 \\odot O 上的动点，则 f(P,\\od... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 T=\\ds{k=1}{100}2^{99+k}\\cdot3^{101-k}，则 T 的末尾数字 0 的个数为__________.\n\nRemember to put your answer on its own line after \"A... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n记 I=120\\displaystyle\\int_0^1\\sqrt{\\df{1-x}{1+x}}\\ \\dd x，则 [I]=__________.\n\nRemember to put your answer on its own line... | 68 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n记 I=\\ds{m=1}{\\infty}\\ds{n=1}{\\infty}\\df{1}{n!n!+m!n!}，则 [10I]=__________.\n\nRemember to put your answer on its own line ... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一个 10\\times10 的数表初始每个位置的数字均为 1，现在做有限次操作，每次把某行或某列的所有数换成相反数. 称非负整数 N 是好的，如果最终数表中 -1 的个数恰为 N. 则好的 N 的个数为__________.\n\nRemember ... | 29 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 A 是正交矩阵（即 AA^T=A^TA=I），且 A 的每个矩阵元均为 \\df{1}{2} 或 -\\df{1}{2}. 则这样的 A 的个数为__________.\n\nRemember to put your answer on its o... | 768 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 10 阶实方阵 A 满足 A^5=0，则 A 的秩的所有可能值之和为__________.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role... | 36 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若实数列 \\{a_n\\} 满足：(1) a_0\\in\\bN_+；(2) 对任意非负整数 i，a_{i+1}=2a_i+1 或 a_{i+1}=\\df{a_i}{a_i+2}；(3) 存在正整数 k，使得 a_k=2014，则称数列 \\{a_... | 60 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设非负整数 a_1,a_2,\\cdots,a_{2022} 的和为 2022. 用 x 表示满足 a_i+a_{i+1}\\geq3 的 i 的个数；用 y 表示满足 a_i\\neq a_{i+1} 的 i 的个数，其中 i\\in\\{1,2,\... | 3032 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 \\{a_i\\}_{i=0}^{\\infty} 是正整数数列，满足 a_0=1，且对任意正整数 n，a_n 是最小的正整数，使得 a_n>1 且\\sigma(a_0a_1\\cdots a_{n-1})\\mid\\sigma(a_0a_1\... | 36 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 \\left[\\df{10^{2022}}{10^{100}-9}\\right] 能被 10^n 整除，则非负整数 n 的最大可能值为__________.\n\nRemember to put your answer on its own l... | 22 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 a_n=\\left(\\df{1+q}{1-q}\\right)\\cdot\\left(\\df{1+q^2}{1-q^2}\\right)^{\\frac{1}{2}}\\cdot\\left(\\df{1+q^4}{1-q^4}\\righ... | 42 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n记 I=\\displaystyle\\int_{\\frac{\\pi}{3}}^{\\frac{\\pi}{2}}\\df{\\dd x}{\\sin x}，则 [100I]=__________.\n\nRemember to put your ... | 54 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数 \\varphi_1(x),\\varphi_2(x) 满足\\varphi_i''(x)-\\ln(x+1)\\varphi_i'(x)+e^x\\varphi_i(x)=0,i=1,2,x>-1.定义 w(x)=\\varphi_1'(x)... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知欧拉常数\\gamma=\\displaystyle\\lim_{n\\rightarrow+\\infty}\\left(1+\\df{1}{2}+\\df{1}{3}+\\cdots+\\df{1}{n}-\\ln n\\right)约等于 0... | 236 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 V 为关于 x 的次数小于或等于 10 的实多项式构成的集合，且 a 为实数. 定义 V 到 V 的线性映射 F 为 F(f)=(x^2f)''+f'-af. 记 I 为使得 F 不是满射的 a 的所有可能值之和，则 [I]=__________.... | 572 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对正整数 n，考虑含有 n 阶元的有限非交换群，记 a_n 为这些群的阶的最小可能值，则 a_1+a_2+a_3+a_4 的值为__________.\n\nRemember to put your answer on its own line aft... | 26 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 n 是正整数，3^n-1 没有 12 以上的质因子，则所有满足条件的 n 的和是__________.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知四面体 PABC，点 A_1 在 \\triangle PBC 内，满足 \\triangle A_1BP, \\triangle A_1CP, \\triangle A_1BC 的面积之比为 3:2:1. G 在线段 AA_1 上，直线 PG 交... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若方格 P 的左右两侧分别有 a,b 个方格，上下两侧分别有 c,d 个方格，其中 a,b,c,d 是正整数，且满足 (a-b)(c-d)=0，则称由这些方格所组成的图形为一个\"十字星\".现有一张由 2014 个方格所组成的 38\\times53... | 13483236 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 a_k 为 \\log_2k, \\log_3k, \\cdots, \\log_{2018}k 中整数项的个数，则 \\ds{k=1}{2018}a_k=__________.\n\nRemember to put your answer on... | 4102 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若关于 x 的不等式 (x-2)e^x<ax+1 有且仅有三个不同的整数解，则整数 a 的最小值为__________.\n\nRemember to put your answer on its own line after \"Answer:\".... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若正实数 x,y 满足 x(x+2y)=9，则 x^5y 的最大值为__________.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": ... | 54 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若整数 a,b,c 满足：0\\leq a\\leq 10, 0\\leq b\\leq 10, 0\\leq c\\leq 10，10\\leq a+b+c\\leq 20，则满足条件的有序数组 (a,b,c) 共有__________.\n\nRe... | 891 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n满足下述条件的最小正实数 k：对任意不小于 k 的 4 个互不相同的实数 a,b,c,d，都存在 a,b,c,d 的一个排列 p, q, r, s，使得方程 (x^2+px+q)(x^2+rx+s)=0 有 4 个互不相同的实根.\n\nRemembe... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nn 支球队进行主客场双循环比赛（即每两支球队比赛两场，其中各有一场主场比赛）. 已知每支球队在一周（从周日到周六的七天）内可以进行多场客场比赛，但如果某周内某球队有主场比赛，那么在这一周内不能安排该球队的客场比赛. 若 4 周内能够完成全部比赛，求 n... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n梦境国有 2016 座城市，星程航空公司希望在若干座城市对之间开通单向航线，使得每座城市恰有一班飞离的航线. 求最小的正整数 k，使得该公司无论怎样开通航线，总能将 2016 座城市分成 k 组，满足每组中任意一座城市不可能用不超过 28 次飞行到达这... | 57 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在数列 \\{x_n\\}_{n=1}^{+\\infty} 中，x_1=1，且对任意正整数 n，x_{n+1}=x_n+[\\sqrt{x_n}]，这里 [x] 表示不超过 x 的最大整数. 则满足 x_n\\geq2019 的最小正整数 n 为__... | 95 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 m,n\\in\\bN，且 m,n\\geq2，若 \\df{2^3-1}{2^3+1}\\cdot\\df{3^3-1}{3^3+1}\\cdot\\cdots\\cdot\\df{n^3-1}{n^3+1}=\\df{m^3-1}{m^3+2}... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正整数 a_1,a_2,\\cdots,a_{31},b_1,b_2,\\cdots,b_{31} 满足：(1) a_1<a_2<\\cdots<a_{31}\\leq2015, b_1<b_2<\\cdots<b_{31}\\leq 2015；(2... | 30720 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一项赛事共有 100 位选手参加. 对于任意两位选手 x,y，他们之间恰比赛一次且分出胜负，用 x\\rightarrow y 表示 x 战胜 y. 如果对任意两位选手 x,y，均能找到某个选手序列 u_1, u_2, \\cdots, u_k (k\... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n记 m=7997\\times7999^3-7998\\times7996^3，求 \\sqrt[3]{m}.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 15995 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有 n 个人排成一行，自左至右按 1,2,\\cdots,n 的顺序报数，凡序号为平方数者退出队伍；剩下的人自左至右再按 1,2,3,\\cdots 的顺序重新报数，凡序号为平方数者退出队伍；如此继续. 在此过程中，每个人都将先后从队伍中退出.用 f(... | 1981 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n前 100 个正整数中，能写成两个正整数平方差的数的总和为__________.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user... | 3795 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若在 1,2,3,\\cdots,18 的任意一个排列中，总能找到连续 6 个数之和不小于 m，则实数 m 的最大值为__________.\n\nRemember to put your answer on its own line after \"... | 57 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若长方体的长、宽、高都是自然数，且所有棱长之和等于它的体积，则称此长方体为\"完美长方体\". \"完美长方体''的体积的最大值为__________.\n\nRemember to put your answer on its own line af... | 120 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若正数 x,y 满足 xy=4000，且 (\\lg x-\\lg2)(\\lg y-\\lg2)=\\df{17}{36}，则 x,y 中较大数的整数部分的位数为__________.\n\nRemember to put your answer o... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定一个边长为 2021 的立方体. 有多少种不同的方式可以在立方体边界的某处添加一个单位小立方体，使得新得到的几何体可以分割为若干 1\\times1\\times k 的长方体，其中 k 是某个大于 1 的整数？\n\nRemember to pu... | 13612182 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将正 2017 边形用其 2014 条在内部不相交的对角线划分成 2015 个三角形. 求其中等腰三角形个数的最大可能值.\n\nRemember to put your answer on its own line after \"Answer:\"... | 2010 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有 2022 个格排成一行，A,B 两人在玩游戏. 初始时，在每个左起奇数号格中填入 A、偶数号格中填入 B. 接下来从 A 开始，两人轮流进行以下操作：选择两个写有自己名字的格，这两个格不相邻，且它们之间的所有格都写有另一人的名字，然后将这些格都改为... | 1011 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n某班有 32 名学生，班上有 10 个兴趣小组，每个兴趣小组恰好有 16 名该班学生参加. 对于任意两名学生，将他们两人中恰有一人参加的兴趣小组的个数的平方称为这两名学生的兴趣差. 设 S 是所有 \\text{C}_{32}^2 个两人组的兴趣差之和... | 14080 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设方程 xyz=900 的全部正整数解为 (x_i,y_i,z_i) (1\\leq i\\leq n). 则 \\ds{k=1}{n}(x_k+y_k+z_k)=\\underline{\\qquad\\qquad}.\n\nRemember to ... | 22572 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设集合 S=\\{1,2,\\cdots,2005\\}. 若 S 中任意 n 个两两互质的数中都至少有一个质数，求 n 的最小值.\n\nRemember to put your answer on its own line after \"Answ... | 16 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 I=\\displaystyle\\int_{\\frac{\\pi}{3}}^{\\frac{\\pi}{2}}\\frac{1}{\\sin x}\\ \\dd x，则 [100I]=__________.\n\nRemember to put... | 54 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设地球是一个半径为 1 的球，点 A 在北纬 60 度、东经 30 度，点 B 在南纬 30 度、西经 90 度. 记 A 到 B 的球面距离为 \\theta，则 \\left[100\\cos^2\\theta\\right]=__________... | 42 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知平面直角坐标系中的点集L=\\{(x,y)\\ |\\ x,y\\in\\{1,2,\\cdots,100\\}\\}.设 A 是由平面上若干个凸多边形组成的集合，满足：A 中每个凸多边形的所有顶点均属于 L，并且 L 中的每个点都恰好是 A 中一... | 1667 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n称一个正整数是好的，如果它有三个不同的正因数的和等于 2022. 求最小的好数.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user... | 1344 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最大的正整数 m，使得存在集合 \\{1,2,\\cdots,100\\} 的 m 个子集 A_1, A_2, \\cdots, A_m，满足对任意 1\\leq i<j\\leq m，都有|A_i\\cup A_j|-|A_i\\cap A_j|=... | 100 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一个小镇上共有 2023 位居民，他们成立了一些互不相同的俱乐部 A_1, A_2, \\cdots, A_n，每个俱乐部至少一人，每个人可以在任意个不同的俱乐部内. 已知对于每个俱乐部 A_i，存在一个不在其中的居民 b_i，考虑所有 b_i 未加入... | 2023 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 f(x)=\\ds{i=1}{2013}\\left[\\df{x}{i!}\\right]. 对整数 n，若关于 x 的方程 f(x)=n 有实数解，则称 n 为\"好数\". 求 \\{1,3,5,\\cdots,2013\\} 中好数的个数.... | 587 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知多项式 f(z)=z^3+az^2+bz+c(a,b,c\\in\\bZ) 的所有根的模均为 20 或 15，则这样的多项式的个数为__________.\n\nRemember to put your answer on its own line... | 572 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数 f:\\{1,2,\\cdots,2019\\}\\rightarrow\\{-1,1\\}，满足对任意 1\\leq k\\leq 2019，存在 1\\leq l\\leq 2019，使得\\ds{i:(l-i)(i-k)\\geq0}{}... | 673 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 S 是一个 35 元集合，\\mathcal{F} 是一个由一些 S 到自身的映射组成的集合. 对正整数 k，称 \\mathcal{F} 具有性质 P(k)，如果对任意 x,y\\in S，都存在 \\mathcal{F} 中的 k 个映射f_1... | 595 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在一次数学奥林匹克比赛中，学生获得的分数来自四部分：代数、几何、数论和组合. 任意两名学生在所有四部分中的分数互不相同. 若一组学生能在至少两部分的分数同时按递增的次序排列，则称该组学生是\"好的\". 求最小的正整数 N，使得任意 N 名学生中均存在... | 730 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 a_1,a_2,a_3,a_4,k 是两两不同的正整数，且不小于 80，满足 a_1^2+a_2^2+a_3^2+a_4^2-4k^2 是正整数. 求 (a_1^2+a_2^2+a_3^2+a_4^2-4k^2)\\cdot k^2 的最小值.\n... | 7225 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若一个有理数的十进制表示的小数部分有限，则称它是\"短的\". 对一个正整数 m，若正整数 t 满足：存在 c\\in\\{1,2,\\cdots,2017\\}，使得 \\df{10^t-1}{cm} 是短的，且对任意 1\\leq k<t，均有 \... | 807 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最小的正整数 n，使得存在整数 a_1,a_2,\\cdots,a_n，满足二次三项式x^2-2(a_1+a_2+\\cdots+a_n)^2x+(a_1^4+a_2^4+\\cdots+a_n^4+1)至少有一个整根.\n\nRemember to... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n记 S=\\displaystyle\\lim_{n\\rightarrow\\infty}\\int_{1}^{2023}(\\cos(nx+n!))^4\\ \\dd x，求 [S].\n\nRemember to put your answer ... | 758 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 g(x)=\\displaystyle\\int_x^{\\pi}\\df{(\\sin t)^5}{t}\\ \\dd t，记 S=\\displaystyle\\int_0^{\\pi}g(x)\\ \\dd x，求 [100S].\n\nRe... | 106 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设方阵M=\\begin{bmatrix}10&-9&0&0&0&0&0&-9\\\\ -9&10&-9&0&0&0&0&0\\\\ 0&-9&10&-9&0&0&0&0\\\\ 0&0&-9&10&-9&0&0&0\\\\ 0&0&0&-9&10&-... | 22 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n矩阵微分方程 \\df{\\dd }{\\dd t}M(t)=B(t) 代表每个元素都满足微分方程 \\df{\\dd}{\\dd t}M_{ij}(t)=B_{ij}(t). 考虑矩阵微分方程 \\df{\\dd}{\\dd t}M(t)=AM(t)... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 A 是 5\\times5 的方阵，A 的第一行是 (1,1,1,1,1)，第二行是 (1,-1,1,-1,1). 已知 \\text{tr}(AA^T)=28，求 [\\det A] 的最大可能值.\n\nRemember to put your... | 72 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 n=5，L_n(x)=\\df{1}{n!}\\cdot\\df{\\dd^n}{\\dd x^n}(x^2-1)^n. 记 S=\\displaystyle\\int_{-1}^1|L_n(x)|^2\\ \\dd x，求 [S].\n\nRem... | 186 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设方程(1+2+\\cdots+k)-3(1+2+\\cdots+t)=1的正整数解 (k,t) 按 k+t 从小到大排列为 (k_1,t_1), (k_2,t_2), \\cdots. 求 k_6.\n\nRemember to put your a... | 3196 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 f:\\{1,2,\\cdots,10\\}\\rightarrow\\{1,2,3,4,5\\}，且对一切 k=1, 2, \\cdots, 9，有 |f(k+1)-f(k)|\\geq 3. 则符合条件的函数 f 的个数为________... | 288 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 f(x) 在 (0,+\\infty) 上严格单调递减，对任意 x\\in(0,+\\infty)，均有 f(x)f\\left(f(x)+\\df{2}{x}\\right)=\\df{1}{3}. 记 g(x)=f(x)+4x^2，则函数... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA5. Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that... | 290 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 x_1,x_2,\\cdots,x_{100} 是 1,2,\\cdots,100 的一个排列，则 S=|x_1-x_2|+|x_2-x_3|+\\cdots+|x_{99}-x_{100}|+|x_{100}-x_1| 的最大值为________... | 5000 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有 2018 名学生围成一圈. 称一个同学是优秀的，如果他两边站着的人和自己的性别都不同. 求优秀的男生数与优秀的女生数的平方差的最大值.\n\nRemember to put your answer on its own line after \"A... | 509040 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n正实数 x,y 满足如下条件：存在 a\\in[0,x], b\\in[0,y]，使得a^2+y^2=2,\\quad b^2+x^2=1,\\quad ax+by=1,则 x+y 的最大值为__________.原始的答案是k\\sqrt{m}+n的... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 a_1,a_2,\\cdots,a_{22}\\in[1,2]，求\\df{\\ds{i=1}{22}a_ia_{i+1}}{\\left(\\ds{i=1}{22}a_i\\right)^2}的最大值，其中 a_{23}=a_1.（王正供题）原... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将一枚骰子连续投掷五次，则事件\"五次出现的点数既不全相同，也不两两互异，且从第二次起每一次的点数都不小于前一次的点数\"的概率为__________.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRe... | 167 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若等差数列 \\{a_n\\} 及正整数 m(m\\geq3) 满足：a_1=1,a_m=2，且\\df{1}{a_1a_2}+\\df{1}{a_2a_3}+\\cdots+\\df{1}{a_{m-1}a_m}=3,则 a_1+a_2+\\cdot... | 23 |
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