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\nFind the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid $$ \\frac{x^{2}}{a^{2}}+\\fr... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n(i) A circle of radius $a$ rolls on the inner side of the circumference of a circle of radius $3a$. Find the area contained wi... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe graph of the equation $x^{y}=y^{x}$ in the first quadrant (i.e., the region where $x>0$ and $y>0$) consists of a straight ... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate in closed form\n\n$$\\sum_{k=1}^{n}\\binom{n}{k} k^{2}$$\n\nNote:\n\n$$\\binom{n}{k}=\\frac{n(n-1) \\cdots(n-k+1)}{1 ... | 24 |
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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$$\\lim _{n \\rightarrow \\infty} \\sum_{j=1}^{n^{2}} \\frac{n}{n^{2}+j^{2}}$$\n\nThe original answer is in \\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 $Answer is the answer to the problem.\n\nTwo uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and t... | 27 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIf $x_{1}, \\ldots, x_{n}$ denote the $n$th roots of unity, evaluate\n$$\n\\pi\\left(x_{i}-x_{j}\\right)^{2} \\quad(i<j)\n$$Wh... | -27 |
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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 $Answer is the answer to the problem.\n\nA player throwing a die scores as many points as on the top face of the die and is to play until his score reaches or passes a... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a fixed positive integer $n$ let $x_{1}, x_{2}, \\ldots, x_{n}$ be real numbers satisfying $0 \\leq x_{k} \\leq 1$ for $k=... | 2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the volume of the four-dimensional hypersphere $x^{2}+y^{2}+z^{2}+t^{2}=r^{2}$, and also the hypervolume of its interior ... | 24 |
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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 $Answer is the answer to the problem.\n\n在平面直角坐标系中, 设点集 \\( \\{P_{1}, P_{2}, \\cdots, P_{4 n+1}\\} \\) = \\{(x, y) \\mid x, y \\text{ 为整数, }|x| \\leqslant n,|y| \\leqs... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite \"word... | 8 |
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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 $Answer is the answer to the problem.\n\nFind the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remai... | 27 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn how many ways can the integers from 1 to n be ordered subject to the condition that, except for the first integer on the le... | 32 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \\neq b$. and let $M_{n}$ denote the $2n \\times 2n$ matri... | 27 |
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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 non-self-intersecting closed polygon with $n$ sides. Let its vertices be $P_1$, $P_{2},\\cdots,P_{n}.$ Iet $m$ ot... | 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 $Answer is the answer to the problem.\n\nEvaluate the following limit: \\lim _{n \\rightarrow \\infty} \\int_{0}^{1} \\int_{0}^{1} \\cdots \\int_{0}^{1} \\cos ^{2}\\le... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet R be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x + y + z\\leqslant 1$. Let $w... | 47 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定一个 $2008 \\times 2008$ 的棋盘，棋盘上每个小方格的颜色均不相同。在棋盘的每一个小方格中填入 $C 、 G 、 M 、 O$ 这 4 个字母中的一个, 若棋盘中每一个 $2 \\times 2$ 的小棋盘中都有 $C 、 G 、... | 2046 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $S_{n}=1+2+3+\\cdots+n, n \\in \\boldsymbol{N}^{*}$, 求 $f(n)=\\frac{S_n}{(n^2+32)S_n^{+}1}$的最大值。原始答案是 $\\frac{m}{n}$ 的形式，请给出... | 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\nfor all polygons $P$, where the double integral is taken over $H(P)$. The original answer is in $\\frac{m}{n}$ format, please ... | 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\nAssume $F$ is a finite field with an odd number of elements $m$. Assume $p(x)$ is an irreducible (i.e., non-factorizable) poly... | 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 $Answer is the answer to the problem.\n\nA point P moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edg... | 1 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p$ be a prime number. Let $J$ be the set of all $2 \\times 2$ matrices $\\binom{a b}{c d}$ whose entries are chosen from ... | 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\nThree numbers are chosen independently at random, one from each of the three intervals $[0, L_i](i=1, 2,3)$. If the distributi... | 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\nLet $F(x)$ be a real valued function defined for all real $x$ except for $x=0$ and $x=1$ and satisfying the functional equatio... | 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\nA particle moving on a straight line starts from rest and attains a velocity $v_0$ after traversing a distance $s_0$. If the m... | 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\nDetermine the minimum value of\n$$(r-1)^{2}+\\left(\\frac{s}{r}-1\\right)^{2}+\\left(\\frac{t}{s}-1\\right)^{2}+\\left(\\frac{... | 22 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA sequence of convex polygons $\\{P_n\\}$, $n\\geqslant0$, is defined inductively as follows. $P_0$ is an equilateral triangle... | 10 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $V$ be the region in the cartesian plane consisting of all points $(x, y)$ satisfying the simultaneous conditions $$|x|\\l... | 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\nEvaluate $$\\lim\\limits_{x\\to1^-}\\prod\\limits_{n=0}^\\infty\\left(\\frac{1+x^{n+1}}{1+x^n}\\right)^{x^n}.$$ Please provide... | 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\nLet $S={(a, b) \\mid a=1,2,..., n, b=1,2,3}$. A rook tour of $S$ is a polygonal path made up of line segments connecting point... | 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 $Answer is the answer to the problem.\n\nFour points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they... | 47 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each continuous function $f:[0,1]\\to\\mathbb{R}$, let $I(f)=$ $\\int _{0}^{1}x^{2}f( x)$ $dx$ and $J( x) = \\int _0^{1}x\... | 17 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $k$ be an integer greater than 1. Suppose $a_0>0$, and define $$a_{n+1}=a_n+\\frac{1}{\\sqrt[k]{a_n}}$$ for $n>0.$ Evaluat... | 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 $Answer is the answer to the problem.\n\nWhat is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1? The original answer i... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设实数 $x_{1}, x_{2}, \\cdots, x_{22}$ 满足对任意 $1 \\leqslant i \\leqslant$ 22, 有 $2^{i-1} \\leqslant x_{i} \\leqslant 2^{i}$ 。 求\n$... | 16769021 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定整数 $n(n \\geqslant 3)$ ，设 $A_{1}, A_{2}, \\cdots, A_{2 n}$是集合 $\\{1,2, \\cdots, n\\}$ 的两两不同的非空子集，记 $A_{2 n+1}=A_{1}$. 求 当$n... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$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 $x_{j}$ ... | 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\n定义 $n$ 元整数组的一次变换为 \n$$ \n\\begin{aligned} \n& \\left(a_{1}, a_{2}, \\cdots, a_{n-1}, a_{n}\\right) \\\\ \n& \\rightarrow\\... | 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将关于 $x$ 的多项式 $f(x)=1-x+x^{2}-x^{3}+\\cdots-x^{19}+x^{20}$ 表为关于 $y$ 的多项式 $g(y)=a_{0}+a_{1} y+a_{2} y^{2}+\\cdots$ $+a_{19} y^{1... | 476837158203126 |
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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已知过点 $(0,1)$ 的直线 $l$ 与曲线 $C: y=x+\\frac{1}{x}(x>0)$ 交于两个不同点 $M$ 与 $N$, 求曲线 $C$ 在点 $M 、 N$ 处切线的交点轨迹。请提供这两个交点坐标的所有整数部分之和。\n\nRem... | 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 $p, q(q \\neq 0)$ 是实数, 方程 $x^{2}-p x+q=0$ 有两个实根 $\\alpha, \\beta$,数列 $\\left\\{a_{n}\\right\\}$ 满足 $a_{1}=p, a_{2}=p^{2}-q,... | 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函数 $f(x)=a^{2 x}+3 a^{x}-2(a>0, a \\neq 1)$ 在区间 $x \\in[-1,1]$ 上的最大值为 8, 则它在这个区间上的最小值是 。已知原始答案的形式为$\\frac{m}{n}$，请给出m + n的值。\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已知 $x>0, y>0$, 且满足 \\(\\left\\{\\begin{array}{l}\\cos ^{2}(\\pi x)+2 \\sin (\\pi y)=1 &\\quad①\\\\ \\sin (\\pi x)+\\sin (\\pi ... | 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已知函数 $f(x)=a \\sin x-\\frac{1}{2} \\cos 2 x+a-\\frac{3}{a}+\\frac{1}{2}, a \\in \\mathbf{R}, a \\neq 0$.\\n（1）若对任意 $x \\in \\m... | 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复平面上，非零复数 $z_{1}, z_{2}$ 在以 $i$ 为圆心， 1 为半径的圆上， $\\overline{z_{1}} z_{2}$ 的实部为零， $z_{1}$ 的辐角主值为 $\\frac{\\pi}{6}$ ，则 $z_{2}=$。请... | 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在 $\\triangle A B C$ 中, $A B=1, A C=2, \\cos B=2 \\sin C$, 则 $B C$ 的长为。原始答案的格式为 $\\frac{\\sqrt{m}+\\sqrt{n}}{k}$，请给出 m + n + k... | 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设 \\(\\alpha, \\beta \\in \\mathbf{R}\\), 若 \\(\\tan (\\alpha+\\beta)=2, \\tan (\\alpha+2 \\beta)=3\\), 则 \\(\\tan \\alpha\\) ... | 22 |
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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 $r$ ，甲、乙两人玩如下的游戏. 黑板上写着一个含有三个绝对值的算式： $$ S=|\\square-\\square|+|\\square-\\square|+|\\square-\\square|, $$ 其中 6 个空格 \"$\\s... | 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已知数列 $\\left\\{a_{n}\\right\\}$ 的各项均为非负实数, 且满足：对任意整数 $n \\geq 2$, 均有 $a_{n+1}=a_{n}-a_{n-1}+n$. 若 $a_{2} a_{2022}=1$, 求 $a_{1}... | 6076 |
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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若正数 $a, b$ 满足 $\\log _{2} a+\\log _{4} b=8, \\log _{4} a+\\log _{8} b=2$, 则 $\\log _{8} a+\\log _{2} b$ 的值为。请给出原始答案中分子与分母绝对值之和... | 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\n设 $a$ 为实数. 若存在实数 $t$, 使得 $\\frac{a-\\mathrm{i}}{t-\\mathrm{i}}+\\mathrm{i}$ 为实数 ( $\\mathrm{i}$为虚数单位), 则 $a$ 的取值范围是 $\\qquad$.... | -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设函数 $f(x)=\\frac{x^{2}+x+16}{x}(2 \\leq x \\leq a)$, 其中实数 $a>2$. 若 $f(x)$ 的值域为 $[9,11]$, 则 $a$ 的取值范围是一个区间$[p, q]$. 请给出$p + q$的... | 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\n不等式 $\\frac{20}{x-9}>\\frac{22}{x-11}$ 的解集为 $\\qquad$，标准答案格式为$(-\\infty,a) \\cup(b,c)$请给出解集的区间端点a+b+c和的值。\n\nRemember to put y... | 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设正数 $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}$ 满足: $a_{1}, a_{2}, a_{3}$ 成公差为 $b_{1}$ 的等差数列, $b_{1}, b_{2}, b_{3}$ 成公比为 $a_{1}... | 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一枚不均匀的硬币，若随机抛掷它两次均得到正面的概率为 \\frac{1}{2} ，则随机抛掷它两次得到正面、反面各一次的概率的近似整数值是多少？\n\nRemember to put your answer on its own line after ... | 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\n已知复数 $z$ 满足 $|z|=1$, 且 $\\operatorname{Re} \\frac{z+1}{\\bar{z}+1}=\\frac{1}{3}$, 请给出 \\operatorname{Re} \\frac{Z}{\\bar{z}} 的... | 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\n若正四棱雉 $P-A B C D$ 的各条棱长均相等, $M$ 为棱 $A B$ 的中点, 则异面直线 $B P$ 与 $C M$ 所成的角的余弦值为 \\qquad。已知原始答案为 \\frac{\\sqrt{m}}{n} 的格式，请给出 m + n... | 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给定正实数 $a, b, a<b$. 设 $x_{1}, x_{2}, \\cdots, x_{2022} \\in[a, b]$,求\n$$\n\\frac{\\left|x_{1}-x_{2}\\right|+\\left|x_{2}-x_{3}\... | 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求出所有满足下面要求的不小于 1 的实数 $t$ ：对任意 $a, b \\in[-1, t]$, 总存在 $c, d \\in[-1, t]$, 使得 $(a+c)(b+d)=1$. 答案标准形式为\\frac{a}{b},请给出满足条件的 $t$ ... | 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方程 $\\sin x=\\cos 2 x$ 的最小的 20 个正实数解之和的原始答案是 $m \\pi$ 的形式，请给出 $m$ 的值。\n\nRemember to put your answer on its own line after \"A... | 130 |
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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 $a=1+10^{-4}$ . 在 $2023 \\times 2023$ 的方格表的每个小方格中填入区间 $[1, a]$ 中的一个实数. 设第 $i$ 行的总和为 $x_{i}$ ，第 $i$ 列的总和为 $y_{i}$ ， $1 \\leq ... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $F_{1}, F_{2}$ 为椭圆 $\\Omega$ 的焦点, 在 $\\Omega$ 上取一点 $P$ (异于长轴端点), 记 $O$ 为 $\\Delta P F_{1} F_{2}$ 的外心, 若 $\\overrightarrow{P ... | 10 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设复数 $z$ 满足 $|z|=1$ ，使得关于 $x$ 的方程 $z x^{2}+2 \\bar{z} x+2=0$ 有实根，则这样的复数 $z$ 的和为以分数形式表示，请给出分子与分母的和。\n\nRemember to put your answ... | -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 $Answer is the answer to the problem.\n\n$[2 \\pi-6,4-\\pi]$\n\n设 $f(x)$ 是定义在 $\\mathbf{R}$ 上的以 2 为周期的偶函数, 在区间 $[1,2]$ 上严格递减, 且满足 $f(\\pi)=1, f(2 \\pi)=0$, 则不等式组 $\\le... | 2 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系 $x O y$ 中, 圆 $\\Omega$ 与抛物线 $\\Gamma: y^{2}=4 x$ 恰有一个公共点, 且圆 $\\Omega$ 与 $x$轴相切于 $\\Gamma$ 的焦点 $F$ 。已知原始答案为 \\( \\fra... | 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称一个复数列 $\\left\\{z_{n}\\right\\}$ 是\"有趣的\"，若 $\\left|z_{1}\\right|=1$ ，且对任意正整数 $n$ ，均有 $4 z_{n+1}^{2}+$ $2 z_{n} z_{n+1}+z_{n}... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系中，设$\\vec{e}$是一个单位向量，向量$\\vec{a}$满足$\\vec{a} \\cdot \\vec{e}=2$，并且$|\\vec{a}|^{2} \\leqslant 5|\\vec{a}+t \\vec{e}|$对任... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $a>0$, 函数 $f(x)=x+\\frac{100}{x}$ 在区间 $(0, a]$ 上的最小值为 $m_{1}$, 在区间 $[a,+\\infty)$ 上的最小值为 $m_{2}$, 若 $m_{1} m_{2}=2020$, 则 $a... | 101 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知集合 $A=\\{1,5\\}, B=\\left\\{x^{2}-3 x+1, x^{2}-4 x+5\\right\\}$ ，若集合 $A \\cup B$ 的元素个数是 3 ，请给出所有可能的实数 $x$ 的和。\n\nRemember to... | 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 $\\triangle A B C$ 中, $\\cos A, \\sin A, \\tan B$ 依次构成以 $\\frac{3}{4}$ 为公比的等比数列。请给出 $\\cot C$ 的值，其原始答案为 \\frac{m}{n} 格式，请给出 ... | 43 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, $a=B C, b=C A, c=A B$. 若 $a^{2}+b^{2}=6, \\cos (A-B) \\cos C=\\frac{2}{3}$,求 $\\triangle A B C$ 的外接圆半径... | 23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $a, b \\in \\mathbf{R}$ .若关于 $z$ 的方程 \n$$ \n\\left(z^{2}+a z+b\\right)\\left(z^{2}+a z+2 b\\right)=0 \n$$ \n有 4 个互不相等的复数... | 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设正四棱锥 $P-A B C D$ 的底面边长与高相等, 点 $G$ 为侧面 $\\triangle P B C$ 的重心,则直线 $A G$ 与底面 $A B C D$ 所成角的正弦值的原始答案为 \\(\\frac{\\sqrt{m}}{n}\\)... | 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复数 $z_{1}, z_{2}, \\cdots, z_{100}$ 满足: $z_{1}=3+2 \\mathrm{i}, z_{n+1}=\\overline{z_{n}} \\cdot \\mathrm{i}^{n}(n=1,2, \\cdot... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $i=\\sqrt{-1}$ 为虚数单位, 则 $i+2 i^{2}+3 i^{3}+\\cdots+2013 i^{2013}$ 的实部与虚部的和为多少。\n\nRemember to put your answer on its own lin... | 2013 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知正三棱锥 $P-A B C$ 的底面边长为 1, 高为 $\\sqrt{2}$, 则其内切球半径是一个分数，形式为 \\frac{m}{n}。请给出 m + n 的值。\n\nRemember to put your answer on its o... | 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在 $\\triangle A B C$ 中, 已知 $\\overrightarrow{A B} \\cdot \\overrightarrow{A C}+2 \\overrightarrow{B A} \\cdot \\overrightarrow... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $P$ 为一圆锥的顶点, $A, B, C$ 是其底面圆周上的三点, 满足 $\\angle A B C=90^{\\circ}$, $M$ 为 $A P$ 的中点. 若 $A B=1, A C=2, A P=\\sqrt{2}$, 则二面角 $M... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, 已知 $\\overrightarrow{A B} \\cdot \\overrightarrow{A C}+2 \\overrightarrow{B A} \\cdot \\overrightarrow... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知正四棱锥 $V-A B C D$ 的高等于 $A B$ 长度的一半， $M$ 是侧棱 $V B$ 的中点， $N$ 是侧棱 $V D$ 上的点，满足 $D N=2 V N$ ，则异面直线 $A M, B N$ 所成角的余弦值的原始答案是 $\\fr... | 22 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $x_{1}, x_{2}, x_{3}$ 是非负实数, 满足 $x_{1}+x_{2}+x_{3}=1$, 求 $(x_{1}+3 x_{2}+5 x_{3})(x_{1}+\\frac{x_{2}}{3}+\\frac{x_{3}}{5})$的... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系 $x O y$ 中, 椭圆 $C$ 的方程为 \\frac{x^{2}}{9}+\\frac{y^{2}}{10}=1, F$ 为 $C$ 的上焦点, $A$为 $C$ 的右顶点, $P$ 是 $C$ 上位于第一象限内的动点, 则四边... | 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设两个严格递增的正整数数列 $\\left\\{a_{n}\\right\\},\\left\\{b_{n}\\right\\}$ 满足: $a_{10}=b_{10}<2017$ ，对任意正整数 $n$ ，有 $a_{n+2}=a_{n+1}+a_{... | 33 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设数列 \\left\\{a_{n}\\right\\} 定义为 a_{1}=1 ，\\n\\na_{n+1}= \\begin{cases}a_{n}+n, & a_{n} \\leqslant n, \\\\ a_{n}-n, & a_{n}>n,... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $x_{1}, x_{2}, x_{3}$ 是非负实数，满足 $x_{1}+x_{2}+x_{3}=1$ ，求 $\\left(x_{1}+3 x_{2}+5 x_{3}\\right)\\left(x_{1}+\\frac{x_{2}}{3}+\... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系 $x O y$ 中, 粗圆 $C$ 的方程为 $\\frac{x^{2}}{9}+\\frac{y^{2}}{10}=1, F$ 为 $C$ 的上焦点, $A$ 为 $C$ 的右顶点, $P$ 是 $C$ 上位于第一象限内的动点, 则... | 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正三棱锥 $P-A B C$ 中, $A B=1, A P=2$, 过 $A B$ 的平面 $\\alpha$ 将其体积平分, 则棱 $P C$ 与平面 $\\alpha$ 所成角的余弦值为 \\frac{m\\sqrt{k}}{n} 的形式，请给出 ... | 18 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设两个严格递增的正整数数列 $\\left\\{a_{n}\\right\\},\\left\\{b_{n}\\right\\}$ 满足: $a_{10}=b_{10}<2017$, 对任意正整数 $n$ ，有 $a_{n+2}=a_{n+1}+a_{... | 33 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all rational numbers $a$ for which the matrix\n$$\n\\left(\\begin{array}{cccc}\na & -a & -1 & 0 \\\\\na & -a & 0 & -... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $Q$ 为有理数集, $Z$ 为整数集. 在坐标平面上，对正整数 $m$ ，定义点集 \n$$ \nA_{m}=\\left\\{(x, y) \\mid x, y \\in \\mathbf{Q}, x y \\neq 0, \\frac{x... | 2021 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定整数 $n \\geqslant 2$ 。设 $x_{1}, x_{2}, \\cdots, x_{n}$ 为任意实数. 求\n$$ \n2 \\sum_{1 \\leqslant i<j \\leqslant n}\\left[x_{i} x_... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求所有的正整数对 $(x, y)$ ，满足 $x^{y}=y^{x^{-y}}$ 。请给出所有满足条件的整数对中元素的和。\n\nRemember to put your answer on its own line after \"Answer:\"... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe Imomi archipelago consists of $n \\geqslant 2$ islands. Between each pair of distinct islands is a unique ferry line that ... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a_{1}, a_{2}, \\\\ldots, a_{n}, b_{1}, b_{2}, \\\\ldots, b_{n}$ be $2 n$ positive integers such that the $n+1$ products $... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求实数 $a$ 的取值范围，使得对任意实数 $x$ 和任意 $\\theta \\in\\left[0, \\frac{\\pi}{2}\\right]$ ，恒有$$(x+3+2 \\sin \\theta \\cos \\theta)^{2}+(x+... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知复数 $z=1-\\sin \\theta+i \\cos \\theta\\left(\\frac{\\pi}{2}<\\theta<\\pi\\right)$, 求 $z$ 的共轭复数 $\\bar{z}$ 的辐角主值为$\\frac{a\\p... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $\\theta=\\arctan \\frac{5}{12}$, 那么复数 $z=\\frac{\\cos 2 \\theta+i \\sin 2 \\theta}{239+i}$ 的辐角主值是。原始答案为 $\\frac{m\\pi}{n}$... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n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 $P$ 的横坐标是 $\\qquad$. ... | 69 |
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