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的所有数字之和。\n\nRemember to put your answer on its own line after \"Answer:\".... | 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已知凸六边形 $P$ 的所有顶点在一个单位正方形的边上, 且所有内角相等. 求 $P$ 的最短边长度的最大可能值。原始答案为$a-\\frac{\\sqrt{m}}{n}$的最简格式，请给出$a+m+n$的值。\n\nRemember to put y... | 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设 $n(n \\geqslant 2)$ 为给定的整数。求最小的实数 $\\lambda$ ，使得对于任意实数 $x_{1}, x_{2}, \\cdots, x_{n} \\in$ $[0,1]$ ，存在 $\\varepsilon_{1}, \\... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定整数 $n \\geqslant 2$ ，设实数 $x_{1}, x_{2}, \\cdots, x_{n}$ 满足:\n(1) $\\sum_{i=1}^{n} x_{i}=0$;\n(2) $\\left|x_{i}\\right| \\leq... | 13 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $0<x 、 y<1$. 求\\(\\frac{x y(1-x-y)}{(x+y)(1-x)(1-y)}\\)的最大值。原始答案为\\(\\frac{m}{n}\\)的格式，请给出m + n的值。\n\nRemember to put your ... | 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求所有的整数 $k$, 使得存在正整数 $a$和 $b$, 满足\n$$\\frac{b+1}{a}+\\frac{a+1}{b}=k.$$ 请给出所有满足条件的整数 $k$ 的和。\n\nRemember to put your answer on ... | 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实数 $a_{1}, a_{2}, \\cdots, a_{n}(n \\geqslant 3)$ 满足\n$a_{1}+a_{2}+\\cdots+a_{n}=0$,\n且 $2 a_{k} \\leqslant a_{k-1}+a_{k+1}(k=... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求所有的实数 $k$, 使得不等式\n$$\na^{3}+b^{3}+c^{3}+d^{3}+1 \\geqslant k(a+b+c+d)\n$$\n对任意 $a 、 b 、 c 、 d \\in[-1,+\\infty)$ 都成立. 原始答案为 \... | 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求所有的正整数 $a$ ，使得对任意正整数 $n \\geqslant 5$ ，均有 $\\left(2^{n}-n^{2}\\right) \\mid\\left(a^{n}-n^{a}\\right)$ 。请给出所有满足条件的正整数$a$的和。\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\n已知凸六边形 $P$ 的所有顶点在一个单位正方形的边上，且所有内角相等. 求 $P$ 的最短边长度的最大可能值。请给出m + n的值，其中原始答案格式为1-\\frac{\\sqrt{m}}{n}。\n\nRemember to put your an... | 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 $x 、 y 、 z \\in(0,1)$ ，满足 $\\sqrt{\\frac{1-x}{y z}}+\\sqrt{\\frac{1-y}{z x}}+\\sqrt{\\frac{1-z}{x y}}=2$. 求 $x y z$ 的最大值. ... | 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\n设 $n$ 为给定的正整数. 求最大的正整数 $k$ , 使得存在三个由非负整数组成的 $k$ 元集 \\begin{aligned} & A=\\left\\{x_{1}, x_{2}, \\cdots, x_{k}\\right\\} ,\\\... | 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 $a 、 b 、 c$ 为两两互质的正整数，且满足 \na^{2}\\left|\\left(b^{3}+c^{3}\\right), b^{2}\\right|\\left(a^{3}+c^{3}\\right), c^{2} \\mid\\... | 39 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $\\alpha 、 \\beta \\in\\left(0, \\frac{\\pi}{2}\\right)$. 求\n$$ \nA=\\frac{\\left(1-\\sqrt{\\tan \\frac{\\alpha}{2} \\cdot ... | 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已知复数 $z_{1} 、 z_{2}$ 满足 $\\left|z_{1}\\right|<1$, $\\left|z_{2}\\right|<1, z_{1}+z_{2}$ 为实数，且 $z_{1}+z_{2}+z_{1} z_{2}=0$ 。求 $... | -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已知集合 $A=\\left\\{x \\mid \\log _{a}(a x-1)>1\\right\\}$ 。若 $3 \\in A$, 请给出 a 的取值范围中所有有限端点的和的6倍。\n\nRemember to put your answer... | 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若存在实数 $r$, 使得圆周 $x^{2}+y^{2}=r^{2}$ 上恰好有 $n$ 个整点, 则 $n$ 可以等于多少？请给出所有可能的n值之和。\n\nRemember to put your answer on its own line af... | 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已知存在实数 $r$, 使得圆周 $x^{2}+y^{2}=r^{2}$ 上恰好有 $n$ 个整点, 则 $n$ 可以等于（）。请提供所有满足条件的n值之和。\n\nRemember to put your answer on its own line... | 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已知等差数列 $\\left\\{a_{n}\\right\\}$ 满足: $a_{1}<0, a_{90} \\geqslant 74, a_{200}<200$, 且该数列在区间 $\\left(\\frac{1}{2}, 8\\right)$ 中... | 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已知数列 $\\left\\{a_{n}\\right\\}$ 满足: $a_{n}=\\left[(2+\\sqrt{5})^{n}+\\frac{1}{2^{n}}\\right]\\left(n \\in \\mathbf{N}^{*}\\rig... | 289 |
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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}$. 求 $\\triangle A B C$ 面积的最大值。原始答案为 ... | 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已知在三棱锥 $D-A B C$ 中, $\\angle A C B=\\angle A B D=90^{\\circ}, C A=C B, \\angle B A D=30^{\\circ}$.若点 $C$ 在平面 $A B D$ 上的射影恰好在 $... | 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，则实数 $a$ 的取值范围为 \\qquad. 请给出区间两端点之和的值。\n\nR... | 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过抛物线 $y=x^{2}$ 上两点 $A\\left(1, y_{1}\\right), B\\left(x_{2}, y_{2}\\right)$ 分别作抛物线的切线 $l_{1}, l_{2}, l_{1}$ 与 $l_{2}$交于点 $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\n设 $P(0, a)$ 是 $y$ 轴上异于原点的任意一点，过点 $P$ 且平行于 $x$ 轴的直线与曲线 $y=\\frac{1}{a} \\ln x$ 交于点 $Q$, 曲线 $y=\\frac{1}{a} \\ln x$ 在点 $Q$ 处的切线交... | 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给定正整数 $a, b(a \\leqslant b)$. 数列 $\\left\\{f_{n}\\right\\}$ 满足: \n$$ \nf_{1}=a, f_{2}=b, f_{n+2}=f_{n+1}+f_{n}(n=1,2, \\cdot... | 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 $Answer is the answer to the problem.\n\n$x \\in \\mathbf{R}$, 用 $[x]$ 表示不超过 $x$ 的最大整数, 并用 $\\{x\\}=x-[x]$ 表示小数部分, 已知: $a_{1}=\\sqrt{2}$, $a_{n+1}=\\left[a_{n}\\right]... | 4049 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, 若 $D$ 在 $B C$ 上, $A D$ 平分 $\\angle B A C, A B=A D=3, C D=2$, 求 $\\triangle A B C$ 的周长的两倍。\n\nRemember ... | 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设 $\\overrightarrow{O A}$ 与 $\\overrightarrow{O B}$ 的夹角为 $\\theta,|\\overrightarrow{O A}|=2,|\\overrightarrow{O B}|=1, \\overr... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对 $1,2, \\cdots, 2 n$ 的任意一个排列 $a_{1}, a_{2}, \\cdots, a_{n}, b_{1}, b_{2}, \\cdots, b_{n}$, 定义 $f(a_{1},a_{2},\\cdots,a_{n}, b... | 30 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nn=1时，实数 $x_{i} \\neq 0, i=1,2, \\ldots, n$. 并且有 $\\sum_{i=1}^{n} x_{i}=0$. 求下式的最小值. \n$$ \n\\left.\\sum_{i=1}^{n} x_{i}^{2}\... | 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给定整数 $n>1$. 已知正整数 $m$ 和集合 $\\{1,2, \\ldots, n\\}$ 的不同子集 $A_{1}, A_{2}, \\ldots, A_{m}$ 满足对任意 $1 \\leq i<j \\leq n, A_{1}, A_{2... | 31 |
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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在锐角三角形 $A B C$ 中, $A B$ 上的高 $C E$ 与 $A C$ 上的高 $B D$ 相交于点 $H$, 以 $D E$ 为直径的圆分别交 $A B 、 A C$ 于 $F 、 G$ 两点, $F G$ 与 $A H$ 相交于点 $K... | 216 |
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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将编号为 $1,2, \\cdots, 9$ 的九个小球随机放置在圆周的九个等分点上，每个等分点上各有一个小球。设圆周上所有相邻两球号码之差的绝对值之和为 $S$。求使 $S$ 达到最小值的放法的概率的分子和分母之和。\n\nRemember to p... | 316 |
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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已知植圆 $\\frac{x^{2}}{16}+\\frac{y^{2}}{4}=1$ 的左右焦点分别为 $F_{1}$ 与 $F_{2}$, 点 $P$ 在直线 $l: x-\\sqrt{3} y+8+2 \\sqrt{3}=0$ 上. 当 $\\a... | 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求函数 $y=\\sqrt{x+27}+\\sqrt{13-x}+\\sqrt{x}$ 的最大和最小值。最大值为\\(m\\)，最小值为 \\(n\\sqrt{p}+\\sqrt{q} \\)的形式，请提供m+n+p+q的值。\n\nRemember ... | 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已知函数 $f(x)=a \\sin x-\\frac{1}{2} \\cos 2 x+a-\\frac{3}{a}+\\frac{1}{2}, a \\in \\mathbb{R}$ 且 $a \\neq 0$ 。 (1)若对任意 $x \\in \... | 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已知正三棱雉 $P-A B C$ 底面边长为 1, 高为 $\\sqrt{2}$, 则其内切球半径的原始答案为\\( \\frac{\\sqrt{m}}{n} \\)的格式，请给出m + n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 $a, b \\in \\mathbf{R}_{+}$, 则 满 足 $\\sqrt{x^{2}-\\sqrt{2} a x+a^{2}}+$ $\\sqrt{x^{2}-\\sqrt{2} b x+b^{2}} \\leqslant \\sqrt... | 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_{0} \\in[0,1)$. 数列 $\\left\\{x_{k}\\right\\}$ 满足: 若 $x_{n-1}<\\frac{1}{2}$, 则 $x_{n}=2 x_{n-1}$, 若 $x_{n-1} \\geq \\fr... | 2022 |
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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 $\\frac{z}{2}$ 与 $\\frac{2}{z}$ 的实部和虚部均属于 $[-1,1]$, 则 $z$ 在复平面上形成轨迹的面积为原始答案为$a - b\\pi$的形式，请给出a + b的值。\n\nRemembe... | 14 |
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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已知凸四边形 $A B C D$ 满足: $A B=1, B C=2, C D=4, D A=3$, 则其内切圆半径取值范围为 $\\qquad$. 该区间端点的和形如\\frac{a\\sqrt{b}+c\\sqrt{d}}{f}，请给出a+b+c+... | 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已知正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的棱长为 1, 以顶点 $A$ 为球心， $\\frac{2 \\sqrt{3}}{3}$ 为半径作一个球，则球面与正方体的表面相交所得到的曲线的长等于原始答案的形式为 $... | 14 |
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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已知正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的棱长为 1, 以顶点 $A$ 为球心， $\\frac{2 \\sqrt{3}}{3}$ 为半径作一个球，则球面与正方体的表面相交所得到的曲线的长等于 $\\frac{m... | 14 |
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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解不等式\n$$\n\\log _{2}\\left(x^{12}+3 x^{10}+5 x^{8}+3 x^{6}+1\\right)<1+\\log _{2}\\left(x^{4}+1\\right).\n$$\n求出x的取值范围并给出该区间的端... | 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一种密码锁的密码设置是在正 $n$ 边形 $A_{1} A_{2} \\cdots A_{n}$ 的每个顶点处赋值 0 和 1 两个数中的一个，同时在每个顶点处涂染红、蓝两种颜色之一，使得任意相邻的两个顶点的数字或颜色中至少有一个相同。假设 $n = ... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$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。原始答案为 $-\\frac{m}{n}$ 的形式，请给出 m + n 的值。... | 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 $y=\\left(a \\cos ^{2} x-3\\right) \\sin x$ 的最小值为 -3 ，则实数 $a$ 的取值范围是。请提供该区间的两个端点之和的整数部分。\n\nRemember to put your answer o... | 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如果 $\\cos ^{5} \\theta-\\sin ^{5} \\theta<7\\left(\\sin ^{3} \\theta-\\cos ^{3} \\theta\\right), \\theta \\in[0,2 \\pi)$, 那么 $... | 14 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定 $n$ 个不同实数，其所有全排列组成的集合为 $A_{n}$ ，对于 $\\left(a_{1}, a_{2}, \\cdots, a_{n}\\right) \\in A_{n}$ ，若恰有两个不同的整数 $i, j \\in\\{1,2, \... | 66 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n直线 $x-2 y-1=0$ 与抛物线 $y^{2}=4 x$ 交于 $A, B$ 两点, $C$ 为抛物线上的一点, $\\angle A C B=90^{\\circ}$ ，则点 $C$ 的坐标和为多少？请提供两个可能坐标点的x坐标与y坐标之和。\... | 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正四棱雉 $P-A B C D$ 中, 侧面是边长为 1 的正三角形, $M, N$ 分别是边 $A B, B C$ 的中点, 则异面直线 $M N$ 与 $P C$ 之间的距离是 $\\qquad$。原始答案是\\frac{\\sqrt{m}}{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\n设椭圆 $\\Gamma$ 的两个焦点是 $F_{1}, F_{2}$, 过点 $F_{1}$ 的直线与 $\\Gamma$ 交于点 $P, Q$. 若 $\\left|P F_{2}\\right|=\\left|F_{1} F_{2}\\right... | 15 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设集合 $\\left\\{\\left.\\frac{3}{a}+b \\right\\rvert\\, 1 \\leqslant a \\leqslant b \\leqslant 2\\right\\}$ 中的最大元素与最小元素分别为 $M, m... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系 $x O y$ 中， $F_{1}, F_{2}$ 分别是椭圆 $\\frac{x^{2}}{2}+y^{2}=1$ 的左、右焦点，设不经过焦点 $F_{1}$ 的直线 $l$ 与椭圆交于两个不同的点 $A, B$ ，焦点 $F_{2... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若实数 $a, b, c$ 满足 $2^{a}+4^{b}=2^{c}, 4^{a}+2^{b}=4^{c}$, 求 $c$ 的最小整数值。标准答案格式为$log _{a} b-\\frac{c}{d}$，请提供计算结果 a+b+c+d的值。\n\nR... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求具有下列性质的所有正整数 $k$ ：对任意正整数 $n, 2^{(k-1) n+1}$ 不整除 $\\frac{(k n)!}{n!}$ 。请给出满足条件的所有 $k$ 的和。\n\nRemember to put your answer on it... | 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若实数 $\\alpha, \\beta, \\gamma$ 构成以 2 为公比的等比数列， $\\sin \\alpha, \\sin \\beta, \\sin \\gamma$ 构成等比数列，则 $\\cos \\alpha$ 的值是以\\fra... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的棱长为 $1, \\alpha$ 为过直线 $B D_{1}$ 的平面，则 $\\alpha$ 截该正方体的截面面积的取值范围是 $\\qquad$。答案的标准格式为$\\... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nToday, Ivan the Confessor prefers continuous functions $f:[0,1] \\rightarrow \\mathbb{R}$ satisfying $f(x)+f(y) \\geq|x-y|$ fo... | 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 $H$ 为 $\\triangle A B C$ 的垂心，且满足 $\\overrightarrow{H A}+2 \\overrightarrow{H B}+6 \\overrightarrow{H C}=0$ ，则角 $B=$ \n请给出... | 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设正三棱锥 $P-A B C$ 的底面边长为 1 、高为 $\\sqrt{2}$ ，过底边 $B C$ 作此三棱锥的截面.则截面面积的最小值为 $\\frac{m \\sqrt{n}}{k}$ 的形式，请给出$m + n + k$ 的值。\n\nRem... | 45 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在半径为 1 的实心球中挖掉一个体积最大的圆锥，再将该圆锥重新融成一个圆柱，则圆柱表面积的最小值为 。原始答案的形式为 $\\frac{m \\sqrt[3]{n} \\pi}{p}$，请给出 $m + n+p$ 的值。\n\nRemember to ... | 29 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $x, y, z$ 为正实数, 且满足 $x^{\\lg x} y^{\\lg y} z^{\\lg z}=5, x^{\\lg y z} y^{\\lg z x} z^{\\lg x y}=2$, 则 $x y z$可能取值的乘积是多少 。\n... | 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 $a_{1}=1, a_{n+1}=\\lambda a_{n}^{2}+2\\left(n \\in \\mathbf{N}^{*}\\right)$, 若数列 $\\left\\{a_{n}\\right\\}$ 有上界, 即存在常数 $M$... | 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设函数 $f(x)=2^{x}-2^{-x}$ 的反函数为 $y=f^{-1}(x)$, 则不等式 $\\left|f^{-1}(x-1)\\right|<1$ 的解集为一个区间$(a,b)$，求$a-b$。\n\nRemember to put yo... | 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数列 $\\left\\{a_{n}\\right\\}$ 满足 $a_{1}=1, \\frac{a_{n+1}-a_{n}}{a_{n}}=\\frac{a_{n+2}-a_{n+1}}{a_{n+2}}\\left(n \\in N^{*}\\r... | 2035 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知复数 $z$ 使得 $z-\\frac{4}{z}$ 为纯虚数, 请给出 $|z-1-i|$ 的最小值的整数近似值。\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在 $\\triangle A B C$ 中, 已知 $\\overrightarrow{A B} \\cdot \\overrightarrow{A C}=2 \\overrightarrow{B C} \\cdot \\overrightarrow... | 23 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, $A, B, C$ 的对边分别为 $a, b, c$. 已知 $2 \\sin ^{2} \\frac{A+B}{2}=1+\\cos 2 C$.若 $c^{2}=2 b^{2}-2 a^{2}$, 求 ... | 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已知两点 $A(-2,0), B(2,0)$, 动点 $P$ 在 $y$ 轴上的射影是 $H$, 且 $\\overrightarrow{P A} \\cdot \\overrightarrow{P B}=$ $2|\\overrightarrow{P... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n\n抛物线 $y=\\frac{1}{2} x^{2}$ 与直线 $l: y=x+4$ 所围成区域中有一个矩形 $A B C D$, 且点 $A, B$ 在抛物线上, 点 $D$ 在直线 $l$ 上, 其中点 $B$ 在 $y$ 轴右侧, 且 $|A ... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $G(V, E)$ 是连通的简单图. 称 $E$ 的子集 $H$ 为 $G$ 的\"偶度子图\"，如果对任何 $x \\in V, H$ 中共有偶数条边以 $x$ 为顶点. 设 $|V|=v,|E|=e$ ，问： $G$ 有多少个\"偶度子图\"？... | 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给定正整数 $n$, 求最大的正整数 $k$, 使得如下命题成立: 对每个 $1 \\leq i \\leq 2 n$,设 $A_{i}$ 是若干个相邻的整数构成的集合（即每个 $A_{i}$ 都形如 $\\{a+1, a+2, \\cdots, a+... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n = 10$. There are $n$ points in the plane, no three of them collinear. Each day, Tom erases one of the points, until the... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 \\triangle A B C 为椭圆 \\frac{x^{2}}{9}+\\frac{y^{2}}{4}=1 的内接三角形，且 A B 过点 P(1,0) ，则 \\triangle A B C 的面积的最大值为 \\frac{m\\sqrt... | 21 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求 $\\lim\\limits _{n \\rightarrow+\\infty} \\sum_{k=0}^{n-1} \\frac{120}{\\sqrt{n^{2}+k n}}$. 原始答案为 $m(\\sqrt{n}-q)$ 的形式，请给出m+... | 243 |
[
{
"content": "Solve the following math problem step by step. 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 $f(x)=x^{3}-3 x^{2}-t x-1, x \\in R$ 有重根, 求 $t$ 的所有可能的值。答案为一个整数，基于原始答案$t=-3$ 或者 $\\frac{m}{n}$，请给出-3 + m + n的值。\n\nRemembe... | 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二阶实矩阵 $X$ 满足 $X^{3}=I_{2}$, 请提供 $\text{tr}(X)$ 的所有可能整数解之和。\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设正四棱柱 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的底面 $A B C D$ 是单位正方形, 如果二面角 $A_{1}-B D-C_{1}$ 的大小是 $\\frac{\\pi}{3}$, 则 $A A_{1}=$ 的原始... | 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已知正四棱雉 $V-A B C D$ 的高等于 $A B$ 长度的一半， $M$ 是侧棱 $V B$ 的中点， $N$ 是侧棱 $V D$ 上点，满足 $D N=2 V N$ ，则异面直线 $A M, B N$ 所成角的余弦值为\\(\\frac{\\... | 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设实数 $a$ 满足 $a<9 a^{3}-11 a<|a|$, 则 $a$ 的取值范围是一个区间，区间的两端点和形如\\frac{a\\sqrt{b}+c\\sqrt{d}}{e}，请给出a+b+c+d+e\n\nRemember to put yo... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n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双曲线 $C$ 的方程为 $x^{2}-\\frac{y^{2}}{3}=1$ ，左、右焦点分别为 $F_{1} 、 F_{2}$ 。过点 $F_{2}$ 作一直线与双曲线 $C$ 的右半支交于点 $P, Q$, 使得 $\\angle F_{1} P... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系 $x O y$ 中, 圆 $C_{1}: x^{2}+y^{2}-a=0$ 关于直线 $l$ 对称的圆为 $C_{2}: x^{2}+y^{2}+2 x-2 a y+3=0$, 请给出直线 $l$ 的方程系数之和。\n\nRememb... | 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 $\\left\\{a_{n}\\right\\}$ 是各项均为正数的等比数列，且 $a_{50}, a_{51}$ 是方程 $100 \\lg ^{2} x=\\lg (100 x)$的两个不同的解。求 $a_{1} a_{2} \\cdots... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$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... | 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... | 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 $f(x)$ 是定义在 $\\mathbf{R}$ 上的函数, 若 $f(x)+x^{2}$ 是奇函数, $f(x)+2^{x}$ 是偶函数,则 $f(1)$ 的值为 \\qquad. 原始答案为 $-\\frac{m}{n}$ 的格式，请给出 $... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$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 $-\\frac{m}{n}$ 的形式，请给出 $m + n$ 的值。\n\nRemembe... | 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 $x O y$ 中, 设 $A B$ 是抛物线 $y^{2}=4 x$ 的过点 $F(1,0)$ 的弦, $\\triangle A O B$ 的外接圆交抛物线于点 $P$ (不同于点 $O, A, B$ ). 若 $P F$ 平分 ... | 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已知圆锥的顶点为 $P$, 底面半径长为 $2$, 高为 $1$. 在圆锥底面上取一点 $Q$, 使得直线 $P Q$与底面所成角不大于 $45^{\\circ}$ ，满足条件的点 $Q$ 所构成的区域的面积为 $m\\pi$ 的形式，请给出 $m$ ... | 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 $F$ 。求圆 $\\Gamma$ 的半径。... | 13 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n平面直角坐标系中, $\\vec{e}$ 是单位向量, 向量 $\\vec{a}$ 满足 $\\vec{a} \\cdot \\vec{e}=2$, 且 $|\\vec{a}|^{2} \\leqslant 5|\\vec{a}+t \\vec{e}|... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设三棱雉 $P-A B C$ 满足 $P A=P B=3, A B=B C=C A=2$, 则该三棱雉的体积的最大值是多少。原始答案的格式为 $\\frac{\\sqrt{m} + n}{k}$ ，请给出 $m + n + k$ 的值。\n\nReme... | 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 $Answer is the answer to the problem.\n\nLet \\(ABC\\) be an equilateral triangle. \\(D\\) and \\(E\\) are midpoints of \\([AB]\\) and \\([AC]\\). The ray \\([DE\\) cu... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven \\( 1 \\le n \\le 455 \\) and \\( n^3 \\equiv 1 \\pmod{455} \\), find the number of solutions.\n\nRemember to put your a... | 9 |
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