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\n令 $A=\\left(\\begin{array}{rrr}-\\frac{2}{3} & 1 & 0 \\\\ -1 & \\frac{4}{3} & 0 \\\\ 2 & -\\frac{4}{3} & \\frac{2}{3}\\end{a... | 50 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n如果实数 $k$ 使得关于 $x, y, z$ 的线性方程组 $\\begin{cases}k x+y+2 z=1 \\\\ 3 x+k y+4 z=1 \\\\ -x+2 y+z=2\\end{cases}$ 无解, 则称其是一个“好实数”. 设所有... | 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已知下列结论成立：在复平面上的多项式 $f(z), g(z)$ 和实数 $r>0$ ，若对 $|z|=r$ ，都有 $|g(z)|<|f(z)|$ ，则在 $\\{z||z|<r\\}$ 中， $f(z)$ 与 $f(z)+g(z)$ 的零点数相等 (... | 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 $Answer is the answer to the problem.\n\n已知二元函数\n$f(x, y)=\\sqrt{4 y^{2}-12 y+10}+\\sqrt{18 x^{2}-18 x+5}+\\sqrt{18 x^{2}+4 y^{2}-12 x y+6 x-4 y+1}$\n的最小值为 $a$ ，求 $\\l... | 80 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA rectangle $ABCD$ is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of $... | 330 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate \n \n$$\\begin{aligned}\n&\\begin{aligned}&~~~~\\quad1\\left(\\frac11+\\frac12+\\frac13+\\frac14+\\frac15+\\frac16+... | 55 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven that $A^4=75600\\times B$. If $A$ and $B$ are positive integers, find the smallest value of $B$.\n\nRemember to put your... | 25725 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $x$ be a fraction between $\\frac{35}{36}$ and $\\frac{91}{183}$. If the denominator of $x$ is $455$ and the numerator and... | 136 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nAn isosceles right triangle is removed from each corner of a square piece of paper so that a rectangle of unequal sides remain... | 56 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$P(x)$ is a polynomial of degree $3n$ such that \n$$\\begin{array}{rcl}P(0)=P(3)=\\cdots&=&P(3n)=2,\\\\P(1)=P(4)=\\cdots&=&... | 4 |
[
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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\nWhat is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an in... | 337 |
[
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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 $Answer is the answer to the problem.\n\n设函数 f(x)=|2-\\log_3x|，正实数 a,b,c 满足 a<b<c，且 f(a)=2f(b)=2f(c). 求 \\df{ac}{b} 的值.\n\nRemember to put your answer on its own line ... | 9 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n定义在 \\bR 上的奇函数 f(x) 同时是以 2 为周期的周期函数，并且在 x\\in(0,1) 时，f(x)=\\ln(x^2-2x+\\frac{3}{2}). 则函数 f(x) 在 [0,10] 上的零点个数为__________.（李纪琛供... | 21 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若存在实数 a 及正整数 n 使得 f(x)=\\cos2x-a\\sin x 在 (0,n\\pi) 内恰有 2022 个零点，则满足条件的正整数 n 的值有__________个.\n\nRemember to put your answer on... | 5 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 f:\\bR\\rightarrow\\bR，若对任意实数 x,y,z 都有\\df{1}{3}f(xy)+\\df{1}{3}f(xz)-f(x)f(yz)\\geq\\df{1}{9},求 \\ds{i=1}{100}[if(i)]，其中... | 1650 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 A 是一个 11 元实数集，令集合 B=\\{uv\\ |\\ u,v\\in A,u\\neq v\\}. 求 B 的元素个数的最小值.\n\nRemember to put your answer on its own line after \... | 17 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对于一个由至少两个正整数构成的集合，若每个元素均与其他至少一个元素不互质，则称该集合为\"芳香集\".设 P(n)=n^2+n+1. 求最小的正整数 b，使得存在非负整数 a，满足集合 \\{P(a+1),P(a+2),\\cdots,P(a+b)\\... | 6 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最小的正整数 m，使得对任意大于 3 的质数 p，都有105\\mid 9^{p^2}-29^p+m.\n\nRemember to put your answer on its own line after \"Answer:\".",
"... | 20 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设数列 \\{a_n\\} 满足 a_0=\\df{1}{2}, a_{n+1}=a_n+\\df{a_n^2}{2012}, n\\geq0. 求所有的整数 k，使得 a_k<1<a_{k+1}.\n\nRemember to put your an... | 2012 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设整数 n\\geq 4，A=\\{a_1,a_2,\\cdots,a_n\\} 是整数集，满足 2023\\in A，且 A 中任意两个元素的差的绝对值是素数. 则这样的集合 A 共有_________个.\n\nRemember to put yo... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n正整数 n 的分拆 P=(x_1,x_2,\\cdots,x_n) 是一个非负整数组，满足 \\ds{k=1}{n}kx_k=n. 正整数 m(m\\leq n) 的分拆 Q=(y_1,y_2,\\cdots,y_m) 称为与 P 相容，如果 y_i\... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 a_1,a_2,\\cdots,a_{100} 是非负整数，同时满足以下条件：(1) 存在正整数 k\\leq100，使得 a_1\\leq a_2\\leq\\cdots\\leq a_k，而当 i>k 时 a_i=0；(2) a_1+a_2+a... | 40940 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 n 为正整数，且 \\ds{k=0}{n}\\df{(-1)^k\\text{C}_n^k}{k^3+9k^2+26k+24}=\\df{1}{312}，则 n=__________.\n\nRemember to put your answer ... | 9 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 9 元集合 A=\\{a+bi\\ |\\ a,b\\in\\{1,2,3\\}\\}，其中 i 是虚数单位. \\alpha=(z_1,z_2,\\cdots,z_9) 是 A 中所有元素的一个排列，满足 |z_1|\\leq|z_2|\\leq... | 8 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知首项系数为 1 的五次多项式 f(x) 满足：f(n)=8n, n=1,2,\\cdots,5，则 f(x) 的一次项系数为__________.\n\nRemember to put your answer on its own line aft... | 282 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在奥林匹克高级中学有 2010 名学生和 100 间教室. 开始时，每名学生任意进入到一间教室. 只要所有学生不都在同一间教室，且存在教室 Y 中的学生数目不少于教室 X 中的学生数目，则教室 X 中的一名学生走到教室 Y 中. 若每分钟只移动一名学生... | 63756 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正整数 n 的正因数为 1=d_1<d_2<\\cdots<d_k=n，其中 k\\geq6. 求所有的 n 使得 n=d_5^2+d_6^2.\n\nRemember to put your answer on its own line after... | 500 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 a_1,a_2,\\cdots,a_{20}\\in\\{1,2,\\cdots,5\\}，b_1, b_2, \\cdots, b_{20} \\in\\{1,2,\\cdots,10\\}. 集合 X=\\{(i,j)\\ |\\ 1\\leq... | 160 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数 f(x) 的定义域为 (0,+\\infty)，且满足f(x)-2xf\\left(\\df{1}{x}\\right)+x^2=0,则 f(x) 的最小值为__________.\n\nRemember to put your answer ... | 1 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将一个 3\\times 3 的正方形的四个角上各去掉一个单位正方形所得到的图形称为\" 十字形\". 在一个 10\\times 11 的棋盘上，最多可以放置多少个互不重叠的十字形？（每个十字形恰好盖住棋盘上的 5 个小方格）\n\nRemember... | 15 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n定义运算 a\\ast b=ab-5\\left[\\df{ab}{5}\\right]. 集合 A=\\{0,1,2,3,4\\}，双射 f:A\\rightarrow A 满足 f(a\\ast b)=f(a)\\ast f(b)，则满足条件的 f... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知数列 \\{a_n\\} 满足 a_{n-1}=a_n+a_{n-2}, n\\geq3. 记前 n 项和为 S_n，若 S_{2018}=2017, S_{2019}=2018，则 S_{2020}=\\underline{\\qquad\\qq... | 1010 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设方程 z^3-2z-4=0 的三个复根为 z_1, z_2, z_3，则它们在复平面上对应的点构成三角形的面积为________.\n\nRemember to put your answer on its own line after \"Answ... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若能从 1,2,\\cdots,100 中取出 k 个两两互质的数，则 k 的最大值为________.\n\nRemember to put your answer on its own line after \"Answer:\".",
"... | 26 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定实数集合 A,B，定义运算A\\otimes B=\\{x\\ |\\ x=ab+a+b,a\\in A,b\\in B\\}.设 A=\\{0,2,4,\\cdots,18\\},B=\\{98,99,100\\}，则 A\\otimes B 中... | 29970 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设数列 a_{n+1}=\\left[\\df{a_n}{2}\\right]+\\left[\\df{a_n}{3}\\right], n=1,2,\\cdots,7，这里 [x] 表示不超过 x 的最大整数. 若 a_8=8，则正整数 a_1 有_... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n问：至多能从集合 \\{1,2,\\cdots,100\\} 中取出多少个数，使得其中任意不同两数的和与积均不为 100 的倍数？\n\nRemember to put your answer on its own line after \"Answe... | 45 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在一个社交网站上有 2022 名用户，其中一些是朋友，朋友关系是相互且永久的. 从现在开始，两名陌生用户只有至少有两个公共朋友，才允许他们成为朋友. 问：初始时至少需要有多少对朋友，才能使得最终任两名用户都成为朋友？\n\nRemember to pu... | 3031 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对于正整数 n，将其各位数字之和记为 s(n)，各位数字之积记为 p(n). 若 s(n)+p(n)=n 成立，就称 n 为巧合数，则所有巧合数的和为__________.\n\nRemember to put your answer on its o... | 531 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n称一个九位数是平衡的，如果数字 1\\sim9 都出现. 将所有平衡的数按递增顺序依次写下，设 S 是所得的数字序列. 求最小的正整数 k，使得 S 的任意两个由 k 个连续数字构成的子列互不相同.\n\nRemember to put your an... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设平面向量 \\textbf{a},\\textbf{b} 满足：|\\textbf{a}|=1, |\\textbf{b}|=2, \\textbf{a}\\bot\\textbf{b}. 点 O, A, B 为平面上的三点，满足：\\overrig... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知抛物线 \\Omega:y=x^2，动线段 AB 在直线 y=\\sqrt{3}x-3 上（B 在 A 右侧），且 |AB|=2\\sqrt{3}. 过 A 作 \\Omega 的切线，取左边的切点为 M. 过 B 作 \\Omega 的切线，取右... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n方程 x^3-9x^2+8x+2=0 有 3 个实根 p,q,r，则 \\df{1}{p^2}+\\df{1}{q^2}+\\df{1}{r^2}=__________.\n\nRemember to put your answer on its ow... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n老师为学生购买纪念品，商店中有三种不同类型的纪念品（同类型的纪念品完全相同），价格分别为 1 元、2 元、4 元. 李老师计划用 101 元，且每种纪念品至少购买一件，则共有__________种不同的购买方案.\n\nRemember to put ... | 600 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设集合 A=\\{1,2,\\cdots,100\\}，函数 f:A\\rightarrow A 满足：(1) 对任意 1\\leq i\\leq 99，|f(i)-f(i+1)|\\leq 1；(2) 对任意 1\\leq i\\leq 100，f(... | 8350 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求所有的正整数 k，使得对任意满足 k(ab+bc+ca)>5(a^2+b^2+c^2) 的正实数 a,b,c，都存在三边长分别为 a,b,c 的三角形.\n\nRemember to put your answer on its own line a... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最大的正实数 k，使得对任意满足 a+b+c=1 的非负实数 a,b,c，均有\\ds{}{}\\df{a}{1+9bc+k(b-c)^2}\\geq\\df{1}{2}.\n\nRemember to put your answer on its ... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最大的正整数 n，使得存在 n 个正整数 x_1<x_2<\\cdots<x_n，满足x_1+x_1x_2+\\cdots+x_1x_2\\cdots x_n=2021.\n\nRemember to put your answer on its o... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在正整数 1,2,\\cdots,20210418 中，有多少个数的数码和是 8？\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "use... | 5100 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n使得 3n+1 与 5n+1 同时成为完全平方数的最小正整数 n=__________.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "... | 16 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n方程 \\df{1}{x}+\\df{1}{y}=\\df{1}{2021} 的正整数解的组数为__________.\n\nRemember to put your answer on its own line after \"Answer:\"."... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若函数 f(x)=x^2-2ax-2a|x-a|+1 有且仅有三个零点，则实数 a 的值为__________.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知数列 \\{a_n\\} 的前 n 项和为 S_n，且 a_1=3, S_n=2a_n+\\df{3}{2}((-1)^n-1). 若 \\{a_n\\} 中存在三项 a_1, a_p, a_q (p,q\\in\\bN_+,1<p<q) 成等差数... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 X=\\{1,2,\\cdots,10\\}，f:X\\rightarrow X 的映射满足① f\\circ f=I_X，其中 f\\circ f 是复合映射，I_X 为 X 上的恒等映射；② |f(i)-i|\\leq 2，对任意 i\\in ... | 401 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nS 是集合 \\{1,2,\\cdots,2023\\} 的子集，满足任意两个元素的平方和不是 9 的倍数，则 |S| 的最大值是__________.（这里 |S| 表示 S 的元素个数）\n\nRemember to put your answer... | 1350 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 f(x)=\\sin\\omega x+\\sin 2x，其中 \\omega\\in\\bN_+, \\omega\\leq2023. 若 f(x)<2 恒成立，则满足题设的常数 \\omega 的个数为__________.\n\nRem... | 1770 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n使得 n^2+2023n 为平方数的正整数 n 的最小值是__________.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user... | 425 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n现有 11 位同学报名博物馆的志愿讲解活动，活动从上午 9 点开始到下午 5 点结束，每小时安排一场公益小讲堂，每场需要 1 位同学为参观的游客提供讲解服务. 为避免同学们劳累，馆方在排班时不会让同一人连续讲解 2 场，并且第一场与最后一场需要两位不同... | 100000010 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 a_1,a_2,\\cdots,a_6;b_1,b_2,\\cdots,b_6;c_1,c_2,\\cdots,c_6 都是 1, 2, \\cdots, 6 的排列. 求 \\ds{i=1}{6}a_ib_ic_i 的最小值.\n\nRememb... | 162 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n某国有 100 个城市，分别编号 1,2,\\cdots,100. 在一些城市之间有道路相连，两个城市之间至多只有一条道路. 称有道路相连的两个城市是相邻的，一位外国游客计划来该国旅行 100 次. 对 1\\leq i\\leq 100，在第 i 次... | 2500 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 z=9+10i（i 为虚数单位），若正整数 n 满足 |z^n|\\leq 2023，则 n 的最大值为__________.\n\nRemember to put your answer on its own line after \"Ans... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设集合 X=\\{1,2,\\cdots,2001\\}. 求最小的正整数 m，使得对 X 的任一 m 元子集 W，都存在 u,v\\in W（u,v 可以相同），满足 u+v 是 2 的幂.\n\nRemember to put your answe... | 999 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求所有的正整数 n，使得存在 k 及大于 3 的整数 n_1, n_2, \\cdots, n_k，满足n=n_1n_2\\cdots n_k=2^{\\frac{1}{2^k}(n_1-1)(n_2-1)\\cdots(n_k-1)}-1.\n\nR... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最大的实数 C，使得不等式\\left(\\ds{k=1}{100}\\df{1}{x_k}\\right)\\left(\\ds{k=1}{100}(x_k^3+2x_k)\\right)-\\left(\\ds{k=1}{100}\\sqrt{x... | 33340000 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正整数 a,b 满足 a<b\\leq100. 若存在正整数 k，使得 ab\\mid a^k+b^k，则称数对 (a,b) 是\"好的\". 求所有好数对的个数.\n\nRemember to put your answer on its own ... | 96 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数列 \\{z_n\\} 满足 |z_1|=1,|z_n+z_{n-1}|=n, n\\geq2. 记 |z_{20}-z_{23}| 和 |z_{20}+z_{23}| 的最大值分别为 m 和 n，则 m-n 的值为__________.\n\n... | 420 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对正整数 n 及实数 x(0\\leq x<n)，定义f(n,x)=(1-\\{x\\})\\cdot\\text{C}_n^{[x]}+\\{x\\}\\cdot\\text{C}_n^{[x]+1},其中 [x] 表示不超过实数 x 的最大整数，\... | 74 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求满足下列条件的大于 5 的最小奇数 a：存在正整数 m_1, n_1, m_2, n_2，使得 a=m_1^2+n_1^2，a^2=m_2^2+n_2^2，且 m_1-n_1=m_2-n_2.\n\nRemember to put your answ... | 261 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n平面上有 2022 个点，满足任意三点不共线. 每个点被染为红色或蓝色，使得每三个（不同的）红点构成的三角形都至少包含一个蓝点. 求红点个数的最大可能值.\n\nRemember to put your answer on its own line a... | 1012 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知数列 \\{a_n\\},\\{b_n\\} 满足 a_1=2, b_1=0, a_2=2,b_2=2，且对 n\\geq3，有 a_n=a_{n-1}a_{n-2}-b_{n-1}b_{n-2}，b_n=a_{n-1}b_{n-2}+a_{n-2... | 64 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设实数 x_1,x_2,\\cdots,x_{2018} 满足 x_1+x_2+\\cdots+x_{2018}=1007, 且 |x_{i+1}-x_i|\\leq1, i=1, 2, \\cdots, 2017. 求 x_2+x_4 的最大值.（赵... | 2014 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将一个 99\\times101 的方格表中若干个小方格涂黑，其余小方格涂白. 如果每个黑格至多和一个黑格相邻（有公边），最多能涂黑多少个小方格？\n\nRemember to put your answer on its own line after... | 5017 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设实数 a,b 满足：集合 A=\\{x\\in\\bR\\ |\\ x^2-10x+a\\leq0\\} 与 B=\\{x\\in\\bR\\ |\\ bx\\leq b^3\\} 的交集为 [4,9]，则 a+b 的值为__________.\n\... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 f(x) 是定义域为 \\bR、最小正周期为 5 的函数. 若函数 g(x)=f(2^x) 在区间 [0,5) 上的零点个数为 25，则 g(x) 在区间 [1,4) 上的零点个数为__________.\n\nRemember to put yo... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n称集合 A 为无和的，是指：若 a,b\\in A，则 a+b\\not\\in A. 设 A_1, A_2 是 U=\\{1,2,\\cdots,999\\} 的两个无和子集，求 |A_1\\cup A_2| 的最大可能值.（吴茁供题）\n\nRem... | 800 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n棋盘上有一只羚羊棋子，它可以从格 (x_1,y_1) 跳到格 (x_2,y_2)，当且仅当 \\{|x_1-x_2|,|y_1-y_2|\\}=\\{3,4\\}. 将 1\\sim10^{12} 任意填在一个 10^6\\times10^6 的方格表... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 a_1<a_2<\\cdots<a_{2024} 是正整数等差数列，b_1<b_2<\\cdots<b_{2024} 是正整数等比数列. 求同时出现在两个数列中的整数个数的最大可能值.\n\nRemember to put your answer ... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若方程 \\log_2\\left(k\\left(x-\\df{1}{2}\\right)\\right)=2\\log_2\\left(x+\\df{1}{2}\\right) 仅有一个实根，则满足条件的 k 的最大值等于__________.\n... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n满足 x^4-x^3-2\\sqrt{5}x^2-7x^2+\\sqrt{5}x+3x+7\\sqrt{5}+17=0 的所有正实数 x 的整数部分之和是__________.\n\nRemember to put your answer on its... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有__________个不超过 2020 的正整数 k，满足对任意的正整数 n，均有3^{(k-1)n+1}\\nmid\\left(\\df{(kn)!}{n!}\\right)^2.\n\nRemember to put your answer o... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最大的正整数 n，使得存在正整数数列 a_1, a_2, \\cdots, a_n，满足对2\\leq k\\leq n-1，有 a_{k+1}=\\df{a_k^2+1}{a_{k-1}+1}-1.\n\nRemember to put your ... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设集合 A=\\{2,3,4,\\cdots,4050\\}，集合 B=\\{(a,b)\\ |\\ \\log_ab+8\\log_ba=6,a\\in A,b\\in A\\}，则集合 B 的元素个数为__________.\n\nRemember... | 68 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知正整数 n 的所有正因数排列为：1=d_1<d_2<d_3<\\cdots，则在 1, 2, 3, \\cdots, 2024 中使得 d_{10}=88 的所有数之和为__________.\n\nRemember to put your ans... | 2376 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n实数 a,b,c 满足 ab+bc+ca=44，求 (a^2+4)(b^2+4)(c^2+4) 的最小值.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 6400 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 3 个整数 a,b,c 满足 a^2+b^2+c^2+3<ab+3b+3c，则这样的有序整数组 (a,b,c) 共有__________组.\n\nRemember to put your answer on its own line after ... | 14 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将正整数 1,2,\\cdots,100 填入 10\\times10 方格表中，每个小方格恰好填 1 个数，要求每行从左到右 10 个数依次递减，对 1\\leq i\\leq 10，记第 i 行的 10 个数之和为 S_i. 设 n\\in\\{1... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n如果关于 x 的三次方程 x^3=ax+a+1 有满足 |x|<1000 的偶数根，求实数 a 的个数.\n\nRemember to put your answer on its own line after \"Answer:\".",
"... | 999 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最小的正整数 m，使得对任意整数 A,B，存在多项式 f(x)，满足 f(-2)=A,f(-3)=B，且 f(x) 的各项系数都是不超过 m 的非负整数.\n\nRemember to put your answer on its own line ... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n集合 \\{0,1,2,\\cdots,2012\\} 中有多少个元素 k，使得 \\text{C}_{2012}^k 是 2012 的倍数？\n\nRemember to put your answer on its own line after \... | 1498 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求所有的正整数 n，使得存在正整数 a,b，满足 a^2+b+3 不能被任何质数的立方整除，且n=\\df{ab+3b+8}{a^2+b+3}.\n\nRemember to put your answer on its own line after ... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在直径为 \\sqrt{2}+1，高为 8 的圆柱形桶内装直径为 1 的球，最多可装__________个.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 36 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n数列 \\{a_n\\} 为：1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, \\cdots. 即先取 a_1=1，接着复制该项粘贴在后面作为 a_2，并添加后继数 2 作为 a_3，再复制所有项 1, 1, ... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知不等式 |ax^2+bx+a|\\leq x 对 x\\in[1,2] 恒成立，则 3a+b 的最大值为__________.\n\nRemember to put your answer on its own line after \"Answe... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 S=\\{1,2,\\cdots,98\\}. 求最小正整数 n，使得 S 的任一 n 元子集中都可以选出 10 个数，无论怎样将这 10 个数均分成两组，总有一组中存在一个数与另外 4 个数都互质，而另一组中总有一个数与另外 4 个数都不互质.\... | 50 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设数列 \\{a_n\\} 满足：(1) a_1 是完全平方数；(2) 对任意正整数 n，a_{n+1} 是使 2^na_1+2^{n-1}a_2+\\cdots+2a_n+a_{n+1} 为完全平方数的最小的正整数. 若存在正整数 s，使得 a_s=... | 31 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 A 是平面内 6 个点组成的集合，记 n(A) 为经过 A 中至少 3 个点的单位圆的个数. 求 n(A) 的最大可能值.\n\nRemember to put your answer on its own line after \"Answer:... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 P 是一个凸 2019 边形，任意三条对角线在形内不交于一点. 称 P 的对角线在形内的交点为\"结点\"，问：最多可以选取多少个结点，使得它们之间不存在一个圈？这里一个圈的任意两个相邻结点必须位于同一条对角线上，且在每条对角线上最多有一个圈上的两... | 2035151 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 p 是质数且 \\df{28^p-1}{2p^2+2p+1} 是整数. 求 2p^2+2p+1 的正因数个数的所有可能值.\n\nRemember to put your answer on its own line after \"Answer:... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n关于 x,y 的方程 x^2+y^2=2023 的整数解所构成的集合 \\{x,y\\} 的个数为__________.\n\nRemember to put your answer on its own line after \"Answer:\".... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 z=\\df{(3i-4)^2(i-\\sqrt{7})^8}{(i-1)^{12}}，则 |z|=__________.\n\nRemember to put your answer on its own line after \"Answer:... | 1600 |
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