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已知点 $P$ 在双曲线 $\\frac{x^{2}}{16}-\\frac{y^{2}}{9}=1$ 上, 并且 $P$ 到这条双曲线的右准线的距离恰是 $P$ 到这条双曲线的两个焦点的距离的等差中项。那么, $P$ 的横坐标的原始答案是 $\\fr... | 69 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定正整数 $n$ 和正数 $M$ 。对于满足条件 $a_{1}^{2}+a_{n+1}^{2} \\leqslant M$ 的所有等差数列 $a_{1}, a_{2}, a_{3}, \\cdots$, 试求 $S=a_{n+1}+a_{n+2}+\... | 30 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n如图3，在直角坐标系 $x O y$中， $\\\\odot O: x^{2}+y^{2}=4$ 与 $x$ 轴的正半轴交于点 $A, \\\\odot A:(x-2)^{2}+y^{2}=r^{2}(r>0)$ 与 \\\\odot O 交于 $B ... | 2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一项考试的可能得分为 $0,1,2, \\cdots, 150$ ，有 100 名考生 $P_{1}, P_{2}, \\cdots, P_{100}$ 考完后依顺时针围成一圈交流成绩，记 $P_{i}$ 的得分为 $a_{i}$ 。每个考生 $P_{... | 1147349 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对任意的正整数 $n$, 记 $D_{n}$ 为 $n$ 的正约数全体， $f_{i}(n)(i=1,2)$ 为集合 $$ F_{i}(n)=\\left\\{a \\in D_{n} \\mid a \\equiv i(\\bmod 4)\\righ... | 2023 |
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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对正整数 $n$ ，定义 $n!=1 \\times 2 \\times \\cdots \\times n$ 。求所有的正整数 $x 、 y$, 使得 $3^{x}+x^{4}=y!+2019$ 。请给出所有解的x + y的值。\n\nRemembe... | 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求最大的正整数 $m$ ，使得存在集合 $\\\\{1,2, \\\\ldots, 100\\\\}$ 的 $m$ 个子集 $A_{1}, A_{2}$ ， $\\\\ldots, A_{m}$ 满足：对任意 $1 \\\\leq i<j \\\\le... | 100 |
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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 $n$ 个人，已知他们中的任意两人至多通电话一次，他们中的任意 $n-2$ 个人之间通电话的次数相等, 都是 $3 k$ 次, 其中 $k$ 是自然数, 求 $n$ 的可能值的和.\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一个球与正四面体的六条棱都相切。若正四面体的棱长为 $a$，则这个球的体积的原始答案是\\frac{\\sqrt{a}\\pi a^b}{c}的格式，请给出a+b+c的值。\n\nRemember to put your answer on its o... | 29 |
[
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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 $S_{n}=1+2+3+\\cdots+n, n \\in \\mathbf{N}$. 求 $f(n)=\\frac{S_{n}}{(n+32) S_{n+1}}$ 的最大值。原始答案为\\frac{m}{n}的格式，请给出m + n的值。\n\... | 51 |
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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若函数 $f(x)=-\\frac{1}{2} x^{2}+\\frac{13}{2}$ 在区间 $[a, b]$ 上的最小值为 $2 a$, 最大值为 $2 b$, 求 $[a, b]$. 标准答案格式为$[a,b]=[m,n]$ 或 $\\Big[... | 40 |
[
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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若复数 $z_{1}, z_{2}$ 满足 $\\left|z_{1}\\right|=2,\\left|z_{2}\\right|=3,3 z_{1}-2 z_{2}=\\frac{3}{2}-i$, 则 $z_{1} \\cdot z_{2}$ 的... | 128 |
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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两个半径都是 $1$ 的球 $O_{1}$ 和球 $O_{2}$ 相切，且均与直二面角 $\\alpha-l-\\beta$ 的两个半平面都相切，另有一个半径为 $r(r<1)$ 的小球 $O$与这个二面角的两个半平面也都相切, 同时,与球 $O_{1... | 10 |
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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若实数 $x$、$y$ 满足 $x^2+y^2=1$,则$\\frac{2xy}{x+y-1}$的最小值是。请给出答案中$a-\\sqrt{b}$的a+b之和。\n\nRemember to put your answer on its own lin... | 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设 $0<\\alpha<\\pi, \\pi<\\beta<2 \\pi$. 若对任意的 $x \\in \\mathbf{R}$ ，都有 $$ \\cos (x+\\alpha)+\\sin (x+\\beta)+\\sqrt{2} \\cos x... | 18 |
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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 $n \\in \\mathbf{N}_{+}$，若 $2^{n} n+1$ 是 3 的整数倍，则 $n$ 被 6 除所得余数构成的集合是_____. 请给出集合中所有元素的和。\n\nRemember to put your answer on... | 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如图1，在第一象限内,矩形 $A B C D$ 的三个顶点 $A 、 B 、 C$分别在函数 $y=\\log _{\\frac{\\sqrt{2}}{2}} x 、 y=x^{\\frac{1}{2}} 、y=\\left(\\frac{\\sqrt... | 33 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n满足 \n \n$\\frac{1-\\sin \\theta+\\cos \\theta}{1-\\sin \\theta-\\cos \\theta}+\\frac{1-\\sin \\theta-\\cos \\theta}{1-\\sin ... | 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已知集合 $M=\\{1,3,5,7,9\\}$. 若非空集合 $A$ 满足: $A$ 中各元素都加 4 后构成 $M$ 的一个子集， $A$ 中各元素都减 4 后也构成 $M$ 的一个子集。请给出集合 $A$ 的元素之和。\n\nRemember t... | 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已知等差数列 $\\left\\{a_{n}\\right\\} 、\\left\\{b_{n}\\right\\}$ 的前 $n$ 项和分别为 $S_{n} 、 T_{n}$, 且对于一切正整数 $n$, 均有 $\\frac{a_{n}}{b_{n... | 43 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 $f(x)=x \\ln x-a x^{2}, a \\in \\mathbf{R}$，设函数 $F(x)=|f(x)|(x \\in[1, \\mathrm{e}])$ 有极小值, $a$ 的取值范围是$\\left(a, \\frac{... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n用 $6$ 根等长的细铁棒焊接成一个正四面体形框架，铁棒的粗细和焊接误差不计。设此框架能容纳得下的最大球的半径为 $R_{1}$,能包容此框架的最小球的半径为 $R_{2}$ 。则 $\\frac{R_{1}}{R_{2}}$等于。原始答案为 $\\f... | 6 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $a$ 为实数, 且对任意 $k \\in[-1,1]$, 当 $x \\in(0,6]$ 时, $6 \\ln x+x^{2}-8 x+a \\leqslant k x$ 恒成立, 则 $a$的最大值是$m-m \\ln m$，求$m$的值。\... | 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设点 $A$ 的坐标为 $(0,3)$, 点 $B, C$ 为圆 $O: x^{2}+y^{2}=25$ 上的两动点, 满足 $\\angle B A C=90^{\\circ}$. $\\triangle A B C$ 面积的最大值为$\\frac... | 71 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知正实数 $x, y$ 满足: $\\frac{1}{x+3 y}+\\frac{1}{2 x+y}=1$, 则 $x+y$ 的最小值是$\\frac{a}{b}(c+d \\sqrt{d})$，求$a+b+c+d$的值。\n\nRemember t... | 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设函数 $f(x)=x-\\ln (a x+2 a+1)+2$. 若对于任意 $x \\geqslant-2$, 均有 $f(x) \\geqslant 0$ 恒成立，请给出实数 $a$ 的取值范围端点之和。\n\nRemember to put yo... | 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已知实数 $x, y$ 满足 $x|x|+\\frac{y|y|}{3}=1$, 则 $|\\sqrt{3} x+y-4|$ 的取值范围为一个区间$[p, q)$，请给出p + q的整数部分的值。\n\nRemember to put your ans... | 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若 $\\triangle A B C$ 的三边 $a, b, c$ 满足 $a^{2}+b^{2}+3 c^{2}=7$, 则 $\\triangle A B C$ 面积的最大值为。原始答案为\\frac{\\sqrt{m}}{n}的格式，请给出m ... | 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给定正整数 $n(n \\geqslant 2)$. 已知 $2 n$ 个正实数 $a_{1}, a_{2}, \\cdots, a_{2 n}$, 满足: $$ \\sum_{k=1}^{n} a_{2 k-1} \\cdot \\sum_{k=1}... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定正整数 $a, b(a \\leqslant b)$. 数列 $\\left\\{f_{n}\\right\\}$ 满足: $$ f_{1}=a, f_{2}=b, f_{n+2}=f_{n+1}+f_{n}(n=1,2, \\cdots) $$ ... | 7 |
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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如图1，已知正方形 ABCD 的边长为 3 $E$为$DC$ 的中点$,AE$ 与 $BD$交于点 $F.$ 则$\n\\overrightarrow{FD}\\cdot\\overrightarrow{DE}$的原始答案为$-\\frac{m}{n}... | 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设等比数列 ${an}$的前$n$项和 为$S_n$ ，且$a_{n+1}=2S_{n}+\\frac{1}{2}(n\\in\\mathbf{Z}_{+}).$\\求数列 ${an}$的通项公式。已知原始答案为$\\frac{a}{b}\\times... | 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已知直线 $l: y=\\sqrt{3} x+4$, 动圆 $O: x^{2}+y^{2}=r^{2}(1<r<2)$, 菱形 $A B C D$ 的一个内角为 $60^{\\circ}$, 顶点 $A, B$ 在直线 $l$ 上, 顶点 $C, D$... | 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某人抛郑一枚硬币，出现正面向上和反面向上的概率均为 $\\frac{1}{2}$. 构造数列 $\\left\\{a_{n}\\right\\}$ ，使\\na_{n}= \\begin{cases}1, & \\text { 第 } n \\text... | 141 |
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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\nLet P be a polynomial, with real coefficients, in three variables and F be a function of two variables such that\n\n$P(ux,uy, ... | 38 |
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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\nFor each positive integer $n$, let $a(n)$ be the number of zeros in the base 3 representation of $n$. For which positive real ... | 25 |
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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\nLet $k$ be a nonnegative integer. Evaluate $$\\sum_{j=0}^k2^{k-j}\\binom{k+j}j.$$ Please provide the value of the expression w... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be an integer with $n\\geq2.$ Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of n... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n;$ that is to say, $p(x)=x^{2n}+a_{2n-1}... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_0=5/2$ and $a_k=a^2_{k-1}-2$ for $k\\geq1.$ Compute $$\\prod\\limits_{k=0}^{\\infty}\\left(1-\\frac{1}{a_k}\\right)$$ i... | 10 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that $f$ is a function from $\\mathbb{R}$ to $\\mathbb{R}$ such that $$f(x)+f\\left(1-\\dfrac{1}{x}\\right)=\\arctan x... | 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\nLet $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k\\leq... | 13 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f:\\mathbb{Z}_{>0}\\to\\... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three li... | 0 |
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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\nFor each permutation $y_1,y_2,\\cdots,y_n$ of these numbers we have $\\sum\\limits_{i=1}^{n-1}y_iy_{i+1}=y_1y_2+y_2y_3+y_3y_4+... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSir Alex plays the following game on a row of $9$ cells. Initially, all cells are empty. In each move, Sir Alex is allowed to ... | 2025 |
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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\nLet $n>1$ be an integer. An $n\\times n\\times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one col... | 91 |
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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\nLet $n$ be a given positive integer. In the Cartesian plane, each lattice point with nonnegative coordinates initially contain... | 10 |
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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\nLet $a_0, a_1, a_2, \\ldots$ be a sequence of real numbers such that $a_0=0, a_1=1$, and for every $n \\geqslant 2$ there exis... | 4033 |
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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\nLet $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so... | 134 |
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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\nVersion 1. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all rea... | 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\nFind all positive integers $n\\geqslant2$ for which there exist $n$ real numbers $a_{1}<\\cdots<a_{n}$ and a real number $r>0$... | 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\nLet $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform ... | 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\nFind all triples of positive integers $(a, b, p)$ with $p$ prime such that $a^p=b!+p$. Please provide the sum of all elements ... | 16 |
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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\nDetermine the maximal length $L$ of a sequence $a_{1}, \\ldots, a_{L}$ of positive integers satisfying both the following prop... | 2025 |
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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\nLet $n$ be a positive integer. We arrange $1+2+\\cdots+n$ circles in a triangle with $n$ rows, such that the $i^{\\text {th }}... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$... | 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\nFind all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$. Please provide the sum ... | 12 |
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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\nFind all ordered pairs $(a,b)$ of positive integers for which the numbers $\\dfrac{a^3b-1}{a+1}$ and $\\dfrac{b^3a+1}{b-1}$ ar... | 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\nSolve the system $$\\left\\{\\begin{array}{l}x^2+y^2-2z^2=2a^2,\\\\x+y+2z=4(a^2+1),\\\\z^2-xy=a^2.\\end{array}\\right.$$ Pleas... | 12 |
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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\nFactor the expression $(b-c)^3+(c-a)^3+(a-b)^3$. The original answer is a product of three differences. Please provide the sum... | 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\nGiven an infinite cone, the measure of its unfolding's angle is equal to $\\alpha$. A curve on the cone is represented on any ... | 179 |
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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\nFind nonzero and nonequal integers $a, b, c$ so that $x( x- a) ( x- b) ( x- c) + 1$ factors into the product of two polynomial... | 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\nSolve in integers the equation $$x+y=x^2-xy+y^2.$$ Please provide the sum of all x and y values from the solutions.\n\nRemembe... | 12 |
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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 $Answer is the answer to the problem.\n\nFind three-digit numbers such that any of its positive integer powers ends with the same three digits and in the same order. P... | 503 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all positive integer solutions of the equation $x^y=y^x\\quad(x\\neq y).$ Please provide the sum of all elements in the s... | 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 $Answer is the answer to the problem.\n\nFind the first three figures after the decimal point in the decimal expression of the number \\(\\frac{0.123456789101112\\ldot... | 239 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn the decimal expression of a positive number, a, all decimals beginning with the third after the decimal point, are deleted ... | 51 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are given segments $AB, CD$ and a point O such that no three of the points $A, B, C, D, O$ are on one straight line. The... | 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\nFind all positive integers n such that $(n − 1)!$ is not divisible by $n^2$. When p is equal to 1,please provide the sum of al... | 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\nDenote by $d(N)$ the number of divisors of N (1 and N are also considered as divisors of N). Find all N such that $N/d(N) = p$... | 59 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nNumber 4 has the following property: when divided by $q^2$, for any q, the remainder is less than $\\frac {q^2}{2}$. Find all ... | 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\nIs there a quadrilateral $ABCD$ of area 1 such that for any point O inside it the area of at least one of the triangles $\\tri... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIs there a real number h such that $[h\\cdot1969^n]$ is not divisible by $[h\\cdot1969^{n-1}]$ for any positive integer n? The... | 3937 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEach pan of a balance has k weights, numbered from 1 to k. The left pan is heavier. It turns out that if we interchange the pl... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA closed non-selfintersecting broken line is drawn on an infinite chessboard along the sides of its squares. There are K black... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider a triangle. The radius of the inscribed circle is equal to $\\frac 43$ ; the lengths of the triangle’s heights are in... | 1871 |
[
{
"content": "Solve the following math problem step by 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 numbers $1, 2, . . . , 1983$ stand at the vertices of a regular 1983-gon. Any of the axes of symmetry of the 1983-gon divi... | 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\nGiven the equation $\\begin{aligned}\\frac{x-49}{50}+\\frac{x-50}{49}=\\frac{49}{x-50}+\\frac{50}{x-49}\\end{aligned}$, if the... | 14801 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose the \"complexity\" of a number is the minimum possible length of a sequence (if any) that begins with 0 and ends with ... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven that in a convex quadrilateral $ABCD$, $\\angle BAC=30^\\circ, \\angle ACD=40^\\circ,\\angle ADB=50^\\circ,\\angle CBD=6... | 290 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven n points on a plane, no three of them lie on a straight line. There is a straight line through each pair of points. When... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA fly is flying inside a regular tetrahedron with side a. The shortest distance for the fly to fly over every face and return ... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $\\triangle A B C$ 及其边 $B C$ 上的一点 $D$ 满足： $A B=2 B D$ ， $A C=3 C D$ ，且以 $A 、 D$ 为焦点可以作一个椭圆 $\\Gamma$ 同时经过 $B 、 C$ 两点，求 $\\G... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知抛物线 $y^{2}=2 p x$ 及定点 $A(a, b), B(-a, 0),\\left(a b \\neq 0, b^{2} \\neq 2 p a\\right). M$ 是抛物线上的点，设直线 $A M, B M$ 与抛物线的另一交点分... | 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给定正整数 $n=3$. 求最大的实数 $\\lambda$, 使得不等式 $a_{n}^{2} \\geqslant \\lambda\\left(a_{1}+a_{2}+\\cdots+a_{n-1}\\right)+2 a_{n}$ 对任何满足 ... | 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设实数 $a$ 满足 $a<9 a^{3}-11 a<|a|$, 则 $a$ 的最小值的$\\sqrt{3}$是 $\\qquad$.\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函数 $y=x+\\sqrt{x^{2}-3 x+2}$ 的最小值\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] | 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设集合 $S$ 是 $\\{0,1, \\cdots, 98\\}$ 的 $m(m \\geqslant 3)$元子集，满足对任意的 $x, y \\in S$ ，均存在 $z \\in$ $S$, 使得 $x+y \\equiv 2 z(\\bmod... | 155 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $f(x)$ 是定义在 $(0,+\nfty)$ 上的减函数。若 $f\\left(2 a^{2}+a+1\\right)<f\\left(3 a^{2}-4 a+1\\right)$成立，请给出 $a$ 的取值范围端点之和的整数部分。\n\nR... | 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若对一切 $\\theta \\in \\mathbf{R}$ ，复数 $z=(a+\\cos \\theta)+(2 a-\\sin \\theta) i$ 的模不超过 2 ，则实数 $a$ 的取值范围的端点值之和。\n\nRemember to p... | 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已知复数 $Z_{1} 、 Z_{2}$ 满足 $\\left|Z_{1}\\right|=2,\\left|Z_{2}\\right|=3$, 若它们所对应的向量的夹角为 $60^{\\circ}$, 则 $7\\sqrt{133}\\left|\\... | 133 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 \n$f(x)=\\lg (x+1)-\\frac{1}{2} \\log _{3} x$. \n(1) 解方程: $f(x)=0$; \n\nRemember to put your answer on its own line afte... | 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 $a_{1}, a_{2}, \\cdots, a_{10}$ 和 $b_{1}, b_{2}, \\cdots, b_{10}$ 均为 $1,2, \\cdots, 10$ 的排列，即集合\n$$\n\\begin{aligned}\n& \\... | 101 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be a positive integer. We arrange $1+2+\\cdots+n$ circles in a triangle with $n$ rows, such that the $i^{\\text {th }}... | 1024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n如果存在 $1,2, \\cdots, n$ 的一个排列 $a_{1}, a_{2}, \\cdots$, $a_{n}$, 使得 $k+a_{k}(k=1,2, \\cdots, n)$ 都是完全平方数，则称 $n$ 为'好数'. 在集合 $\\{1... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系中，设点集\n$\\{P_{1},P_{2},\\cdots,P_{4n+1}\\}$\n$=\\left\\{(x,y)|x,y\\text{ 为整数},|x|\\leq n,|y|\\leq n,xy=0\\right\\}$,\n... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定正整数 $n(n \\geqslant 3)$. 对于 $1,2, \\cdots, n$的任意一个排列 $P=\\left(x_{1}, x_{2}, \\cdots, x_{n}\\right)$, 若 $i<j<k$, 则称 $x_{j}$ ... | 1024 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定素数 $p \\geqslant 5$. 若 $p =7$，求三个连续正整数的乘积模 $p$ 的不同余数的个数。\n\nRemember to put your answer on its own line after \"Answer:\".",... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求最大的实数 $C$ ，使得对任意的正整数 $n$ 和满足 $0=x_{0}<x_{1}<x_{2}<\\cdots<x_{n}=1$ 的数列 $\\left\\{x_{k}\\right\\}$ ，均有$$\\sum_{k=1}^{n} x_{k}^... | 4 |
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