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 $Answer is the answer to the problem.\n\n$A(x)=\\sum_{m=1}^{\\infty} \\sum_{n=0}^{\\infty} \\frac{m!\\cdot x^{m} \\cdot n!}{(m+n+1)!}$, 则 $\\left[100 A\\left(\\frac{2}... | 109 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n正方体旋转群元素最多的两个共轭类的元素个数之和为 $\\qquad$. \n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
... | 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对 $n=5$, 令 $L_{n}=\\frac{1}{n!} \\frac{\\mathrm{d}^{n}}{\\mathrm{~d} x^{n}}\\left(x^{2}-1\\right)^{n}, S=\\int_{-1}^{1}\\left|... | 186 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将方程 $(1+2+\\cdots+k)-3(1+2+\\cdots+t)=1$ 的全体正整数解 $(k, t)$ 按照从小到大的方式排列，得到序列 $\\left\\{\\left(k_{n}, t_{n}\\right)\\right\\}$ ，则... | 3196 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对一个复方阵, 定义 $\\cos A=I+\\sum_{n=1}^{\\infty} \\frac{(-1)^{n}}{(2 n)!} A^{2 n}$ 。记矩阵 $\\cos \\left(\\begin{array}{cc}-5 \\pi & 3... | 20 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求 $\\left|2+2 \\mathrm{e}^{\\frac{2}{5} \\pi i}+\\mathrm{e}^{\\frac{6}{5} \\pi i}\\right|$.\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对于一个实数 $x$, 令 $\\{x\\}=x-[x]$. 记 $S=\\int_{0}^{2024} \\min \\left(\\left\\{\\frac{x}{8}\\right\\},\\left\\{\\frac{x}{4}\\right... | 759 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n令 \n$$ \nS=\\sum_{m=1}^{\\infty} \\sum_{n=1}^{\\infty} \\frac{m^{2} n}{2^{m}\\left(n 2^{m}+m 2^{n}\\right)} \n$$ \n则 $[100... | 200 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知圆锥面 $x^{2}+y^{2}=\\frac{z^{2}}{3}$, 记沿该圆锥面从 $P(-\\sqrt{3}, 3,6)$ 到 $Q(\\sqrt{3}, 0,3)$ 的曲线长度的最小为值为 $I$, 则 $[10 I]=$ $\\qquad... | 60 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$\\frac{4}{x+1}+\\frac{9}{x+2}+\\frac{16}{x+3}=(4 x+5)(2-x)$ 有几个正实数解?\n\nRemember to put your answer on its own line after \"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已知 $z=\\cos \\frac{2}{7} \\pi+\\mathrm{i} \\sin \\frac{2}{7} \\pi$, 求 $\\frac{z}{1+z^{2}}+\\frac{z^{4}}{1+z^{3}}+\\frac{z^{3}}... | -2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系内, $M \\in\\left\\{(x, y) \\left\\lvert\\, \\frac{x^{2}}{200}+\\frac{y^{2}}{8} \\leq 1\\right.\\right\\}, A(2,1)$, 若 $... | 65 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 9 元集合 $A=\\{a+b \\mathrm{i} \\mid a, b \\in\\{1,2,3\\}\\}$, i 是虚数单位. $\\alpha=\\left(z_{1}, z_{2}, \\cdots, z_{9}\\right)$ 是... | 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设集合 $S=\\{1,2,3, \\cdots, 10\\}, S$ 的子集 $A$ 满足\n$$\nA \\cap\\{1,2,3\\} \\neq \\varnothing, \\quad A \\bigcup\\{4,5,6\\} \\neq ... | 888 |
[
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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集合 $A=\\{1,2,3\\}, X=\\{4 x+y \\mid x, y \\in A\\}, Y=\\{4 x-y \\mid x, y \\in A\\}$, 则 $X \\cap Y$的元素之和为 $\\qquad$.\n\nRememb... | 48 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $5 \\times 5$ 矩阵中, 每个元素都为 0 或 1, 且满足：五行的元素之和都相等,但五列的元素之和两两不等。这样的矩阵的个数为 $\\qquad$ \n\nRemember to put your answer on its own ... | 26400 |
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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，双曲线 $\\Gamma: \\frac{x^{2}}{3}-y^{2}=1$ 。对平面内不在 $\\Gamma$ 上的任意一点 $P$ ，记 $\\Omega_{P}$ 为过点 $P$ 且与 $\\Gamma$ 有两个交点的直线的... | 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集合 $A=\\left\\{n \\mid n^{3}<2022<3^{n}, n \\in \\mathbf{Z}\\right\\}$ 的所有元素之和为 $\\qquad$.\n\n\nRemember to put your answer on... | 57 |
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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求具有下述性质的最小正整数 $t$ ：将 $100 \\times 100$ 的方格纸的每个小方格染为某一种颜色，若每一种颜色的小方格数目均不超过104，则存在一个 $1 \\times t$ 或 $t \\times 1$ 的矩形，其中 $t$ 个小... | 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平面直角坐标系 $x O y$ 中, 已知圆 $\\Omega$ 与 $x$ 轴、 $y$ 轴均相切, 圆心在椭圆 $\\Gamma: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0)$ 内，且 $\... | 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若三个正整数 $a, b, c$ 的位数之和为 8, 且组成 $a, b, c$ 的 8 个数码能排列为 2,0,2,4,0,9,0,8 ，则称 $(a, b, c)$ 为 \"幸运数组\"，例如 $(9,8,202400)$ 是一个幸运数组. 满足 ... | 591 |
[
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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若三个正整数 $a, b, c$ 的位数之和为 8, 且组成 $a, b, c$ 的 8 个数码能排列为2, 0,2,4,0,9,0,8, 则称 $(a, b, c)$ 为 \"\"幸运数组\"\"，例如（ $9,8,202400)$ 是一个幸运数组。... | 291 |
[
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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 $2 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}=3$ 的非负整数解共有 $\\qquad$组.\n\n\nRemember to put your answer on... | 174 |
[
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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在直角坐标系 xoy 中，点 $A\\left(x_{1}, y_{1}\\right)$ 和点 $B\\left(x_{2}, y_{2}\\right)$ 的坐标均为一位的正整数。 $O A$ 与 $x$ 轴正方向的夹角大于 $45^{\\circ... | 1985 |
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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在直角坐标系 $x O y$ 中, 点 $A\\left(x_{1}, y_{1}\\right)$ 和点 $B\\left(x_{2}, y_{2}\\right)$ 的坐标均为一位正整数, $O A$ 与 $x$ 轴正方向的夹角大于 $45^{\\... | 1985 |
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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$ 为非负实数，且满足方程 $4^{\\sqrt{5 x+9 y+4 z}}-68 \\times 2^{\\sqrt{5 x+9 y+4 z}}+256=0$ ，那么 $x+y+z$ 的最大值与最小值的乘积等于 $\\qqua... | 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在底面半径为 6 的圆柱内, 有两个半径也为 6 的球面，其球心距为 13 。若作一平面与这二球面相切，且与圆柱面相交成一栯圆，则这个粨圆的长轴长与短轴长之和是 $\\qquad$.\n\n\nRemember to put your answer o... | 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\n已知集合 $U=\\{0,1,2, \\cdots, 2021\\}, S \\subseteq U$, 且 $S$ 中任意两项相加不是 5 的倍数, 求 $S$ 的元素个数最大值.\n\nRemember to put your answer on ... | 810 |
[
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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 $U=\\{0,1,2, \\cdots, 2021\\}, S \\subseteq U$, 且 S 中任意两项相加不是 5 的倍数, 求 S 的元素个数最大值.\n\nRemember to put your answer on its ... | 810 |
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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 $10 \\times 10$ 的数表在初始状态下每个格内均填有数字1，现在对数表进行如下操作：选择某一行或某一列，并将该行（列）中所有格子内的数变为其相反数。对于非负整数 N ，如果可经过有限次操作，使得最终数表中 -1 的个数恰为 N ，则称... | 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记所有2阶实方阵构成的线性空间为 $M_{2}(\\mathbb{R})$ ，已知 $A=\\left(\\begin{array}{cc}3 & 7 \\\\ 10 & 7\\end{array}\\right), B=\\left(\\begin{... | 70 |
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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求六个元素的置换群 $S_{6}$ 中 6 阶元素的个数。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}
] | 240 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n记 $I=\\lim\\limits_{x \\rightarrow+\\infty}\\left(\\sin \\frac{1}{x^{2}}+\\cos \\frac{1}{x}\\right)^{2 x^{2}}$, 则 $[10 I]=$ $\... | 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\n记 $I=\\sum\\limits_{m=1}^{+\\infty} \\sum\\limits_{n=1}^{+\\infty} \\frac{1}{n!n!+m!n!}$, 则 $[10 I]=$ $\\qquad$.\n\nRemember t... | 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记 $\\omega_{n}=\\exp \\left(\\frac{2 \\pi \\sqrt{-1}}{n}\\right)$ 。已知 n 阶傅里叶矩阵定义为 \n$$ \n\\left[\\begin{array}{ccccc} \n1 &... | 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设 V 是所有次数不超过 100 的实系数多项式构成的线性空间。令\n$$\n\\begin{gathered}\nV_{1}=\\left\\{f \\in V \\mid f^{(2 k-1)}(0)=0, k=1,2, \\cdots, 50\\... | 86 |
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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现有 20 盘红灯和 10 盛绿灯。初始时，所有灯都是亮着的。我们每次从所有亮着 的灯中等可能地随机婂灭一用，直到某一种颜色的灯全部被熄灭为止。用 a 表示停止后 仍旧亮着的灯的数目的期望值，则 $[10 a]=$ $\\qquad$\n\nRemem... | 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$f(x)=\\sum_{n=0}^{\\infty} 2^{-n} \\cos (n x)$, 记 $A=\\int_{0}^{2 \\pi} f^{2}(x) \\mathrm{d} x$, 则 [10A] $=$\n\nRemember to p... | 73 |
[
{
"content": "Solve the following math problem step by step. 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 $n=5$, 令 $L_{n}=\\frac{1}{n!} \\frac{\\mathrm{d}^{n}}{\\mathrm{~d} x^{n}}\\left(x^{2}-1\\right)^{n}, S=\\int_{-1}^{1}\\left|... | 186 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n将方程 $(1+2+\\cdots+k)-3(1+2+\\cdots+t)=1$ 的全体正整数解 $(k, t)$ 按照从小到大的方式排列，得到序列 $\\left\\{\\left(k_{n}, t_{n}\\right)\\right\\}$ ，则... | 3196 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$I=\\int_{0}^{\\pi / 4} \\tan ^{101} x \\mathrm{~d} x$ ，则 $\\left[\\frac{1}{3 I}\\right]$\n\nRemember to put your answer on it... | 67 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对一个复方阵, 定义 $\\cos A=I+\\sum_{n=1}^{\\infty} \\frac{(-1)^{n}}{(2 n)!} A^{2 n}$. 记矩阵 $\\cos \\left(\\begin{array}{cc}-5 \\pi & 3... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在透明的球袋中有 1 个黑球和 2 个白球。接下来从袋中摸球，每次摸出一个球，然后放回袋中，并继续往袋中再放入 2 个与摸出球同色的球. 记 $S_{n}$ 为前 n 次中摸出黑球的次数. 已知第 2 次摸出的球是黑球，记 $S_{100}$ 的期望为... | 60 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n5 级方阵 A 满足: 第一行元素均为 1, 第二行元素从左至右依次为 $1,-1,1,-1,1$,且 $\\operatorname{trace}\\left(A A^{\\prime}\\right)=28$. 记 $\\operatorname{... | 72 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n令 $I=\\sqrt{\\frac{2}{\\pi}} \\int_{-\\infty}^{+\\infty} e^{x-10 \\cosh (2 x)} \\mathrm{d} x$, 则 $[|\\ln I|]=$ $\\qquad$.\n\nR... | 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已知圆锥面 $x^{2}+y^{2}=\\frac{z^{2}}{3}$, 记沿该圆锥面从 $P(-\\sqrt{3}, 3,6)$ 到 $Q(\\sqrt{3}, 0,3)$ 的曲线长度的最小为值为 I, 则 $[10 I]=$ $\\qquad$.... | 60 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n令 $A, B, C, D, E, F$ 是三阶实方阵, 且 \n$$ \n\\left(\\begin{array}{ccc} \nI_{3} & A & C \\\\ \n& I_{3} & B \\\\ \n& & I_{3} \n\... | 60 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n点集 $S=\\left\\{(x, y) \\mid x \\leq 5, y \\leq 4, x, y \\in \\mathrm{~N}^{*}\\right\\}$, 则由 S 中的点可组成 $\\qquad$个不同的三角形。\n\nReme... | 1056 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在平面直角坐标系内, $M \\in\\left\\{(x, y) \\left\\lvert\\, \\frac{x^{2}}{200}+\\frac{y^{2}}{8} \\leq 1\\right.\\right\\}, A(2,1)$, 若 $... | 65 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $M=1^{4}+2^{4}+\\cdots+2013^{4}$. 则 $M$ 的个位数字为 $\\qquad$.\n\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若数列 $\\left\\{a_{n}\\right\\}$ 满足\n$$\na_{n}=\\left[(2+\\sqrt{3})^{2^{n}}\\right]\\left(n \\in \\mathbf{Z}_{+}\\right),\n$$\n则... | 53 |
[
{
"content": "Solve the following math problem step by step. 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\n$3|x+1|+2|y-1| \\leqslant 6$.\n则 $2 x-3 y$ 的最大值是 $\\qquad$.\n\n\nRemember to put your answer on its own line a... | 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 $(-x y+2 x+3 y-6)^{6}$ 的展开式中， $x^{4} y^{3}$的系数是 $\\qquad$ （用具体数字作答）\n\n\nRemember to put your answer on its own line after \... | -21600 |
[
{
"content": "Solve the following math problem step by step. 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 $m>0$. 若函数\n$$\nf(x)=x+\\sqrt{100-m x}\n$$\n的最大值为 $g(m)$, 求 $g(m)$ 的最小值.\n\n\nRemember to put your answer on its own line a... | 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 $(1-2 x)^{7}=\\sum_{k=0}^{7} a_{k} x^{k}$. 则\n$2 a_{2}+3 a_{3}+4 a_{4}+5 a_{5}+6 a_{6}+7 a_{7}=$\n\n\nRemember to put your a... | 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在矩形 $A B C D$ 中, $A B=3, A D=4, P$为矩形 $A B C D$ 所在平面上一点，满足 $P A=2$ ， $P C=\\sqrt{21}$. 则 $\\overrightarrow{P B} \\cdot \\overr... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设直线 $y=k x+b$ 与曲线 $y=x^{3}-x$ 有三个不同的交点 $A 、 B 、 C$, 且 $|A B|=|B C|=2$ 。则 $k$ 的值为 $\\qquad$.\n\n\nRemember to put your answer o... | 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对于集合 $\\{x \\mid a \\leqslant x \\leqslant b\\}$ ，我们把 $b-a$ 称为它的长度。设集合 $A=\\{x \\mid a \\leqslant x \\leqslant$ $a+1981\\}, B=... | 983 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定凸 20 边形 $P$ 。用 $P$ 的 17 条在内部不相交的对角线将 $P$ 分割成 18 个三角形。所得图形称为 $P$ 的一个三角剖分图。对 $P$ 的任意一个三角剖分图 $T, P$ 的 20 条边以及添加的 17 条对角线均称为 $T$... | 89 |
[
{
"content": "Solve the following math problem step by step. 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 $S=\\{1,2,3, \\cdots, N\\}$ 的四个500元子集 $A_{1}, A_{2}$, $A_{3}, A_{4}$ 满足：对任意 $x, y \\in S$ ，均存在某个 $i \\in\\{1,2,3,4\\}$ ，使... | 833 |
[
{
"content": "Solve the following math problem step by step. 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 $2 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}=3$ 的非负整数解共有 $\\qquad$组.\n\nRemember to put your answer on i... | 174 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在直角坐标系 $x O y$ 中, 点 $A\\left(x_{1}, y_{1}\\right)$ 和点 $B\\left(x_{2}, y_{2}\\right)$ 的坐标均为一位正整数, $O A$ 与 $x$ 轴正方向的夹角大于 $45^{\\... | 1985 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $x 、 y 、 z$ 为非负实数，且满足方程 $4^{\\sqrt{5 x+9 y+4 z}}-68 \\times 2^{\\sqrt{5 x+9 y+4 z}}+256=0$ ，那么 $x+y+z$ 的最大值与最小值的乘积等于 $\\qqua... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $x, y, z$ 为非负实数，且满足方程 $4^{\\sqrt{5 x+9 y+4 z}}-68 \\times 2^{\\sqrt{5 x+9 y+4 z}}+256=0$ ，那么 $x+y+z$ 的最大值与最小值的乘积等于 $\\qquad$... | 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在底面半径为 6 的圆柱内, 有两个半径也为 6 的球面，其球心距为 13 。若作一平面与这二球面相切，且与圆柱面相交成一栯圆，则这个粨圆的长轴长与短轴长之和是 $\\qquad$.\n\nRemember to put your answer on ... | 25 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知集合 $M=\\{x, y, \\lg (x y)\\}$ 及 $N=\\{0,|x|, y\\}$ ，并且 $M=N$ ，那么 $\\left(x+\\frac{1}{y}\\right)+\\left(x^{2}+\\frac{1}{y^{2}... | -2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设集合 $A=\\{1,2, m\\}$ ，其中 $m$ 为实数。令 $B=\\left\\{a^{2} \\mid a \\in A\\right\\}, C=A \\cup B$ 。若 $C$ 的所有元素之和为 6, 则 $C$ 的所有元素之积为 ... | -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 $5 \\times 5$ 矩阵中, 每个元素都为 0 或 1, 且满足：五行的元素之和都相等,但五列的元素之和两两不等.这样的矩阵的个数为 $\\qquad$ （答案用数值表示）.\n\nRemember to put your answer o... | 26400 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n集合 $A=\\left\\{n \\mid n^{3}<2022<3^{n}, n \\in \\mathbf{Z}\\right\\}$ 的所有元素之和为 $\\qquad$.\n\nRemember to put your answer on i... | 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设 $a_{1}, a_{2}, \\cdots, a_{100}$ 是非负整数，同时满足以下条件：\n（1）存在正整数 $k \\leq 100$ ，使得 $a_{1} \\leq a_{2} \\leq \\cdots \\leq a_{k}$ ，... | 40940 |
[
{
"content": "Solve the following math problem step by step. 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 $\\triangle A B C$ 的三个内角 $A, B, C$ 满足 $\\cos A=\\sin B=2 \\tan \\frac{C}{2}$, 则 $\\sin A+\\cos A$ $+2 \\tan A$ 的值为 $\\qquad$... | 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若正实数 $a, b$ 满足 $a^{\\lg b}=2, a^{\\lg a} \\cdot b^{\\lg b}=5$, 则 $(a b)^{\\lg a b}$ 的值为 $\\qquad$.\n\nRemember to put your ans... | 20 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求具有下述性质的最小正整数 $k$ ：若将 $1,2, \\cdots, k$ 中的每个数任意染为红色或者蓝色，则或者存在 9 个互不相同的红色的数 $x_{1}, x_{2}, \\cdots, x_{9}$ 满足 $x_{1}+x_{2}+\\cd... | 408 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设实数 $a, b$ 满足: 集合 $A=\\left\\{x \\in \\mathbf{R} \\mid x^{2}-10 x+a \\leq 0\\right\\}$ 与 $B=\\left\\{x \\in \\mathbf{R} \\mid ... | 7 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设集合 $A=\\{1,2, \\cdots, 99\\}, X, Y$ 均为 $A$ 的非空子集（允许 $X=Y$ ）。 $X$ 中的最大元与 $Y$ 中的最小元分别记为 $\\max X, \\min Y$ 。求满足 $\\max X>\\min ... | 24 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对正整数 $n$ 及实数 $x(0 \\leqslant x<n)$ ，定义\n$$\nf(n, x)=(1-\\{x\\}) \\cdot\\binom{[x]}{n}+\\{x\\} \\cdot\\binom{[x]+1}{n}\n$$\n其中 ... | 74 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定整数 $n \\geqslant 3$, 设 $a_{1}, a_{2}, \\cdots, a_{2 n}, b_{1}, b_{2}, \\cdots, b_{2 n}$ 是 $4 n$ 个非负实数, 满足 \n$$ \na_{1}+a_{... | 16 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$\\frac{1}{1-\\log _{2} 20}+\\frac{1}{1-\\log _{5} 50}$ 的值为 $\\qquad$.\n\nRemember to put your answer on its own line after \"... | -1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $f(x)=x^{3}+3\\left(x^{2}+x+\\sin \\pi x\\right)$, 则 $f(1-\\pi)+f(\\pi-3)=\\qquad$.\n\nRemember to put your answer on its ow... | -2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设正整数 $a, b, c$ 中任意两个数不互素，且 $a+b+c$ 为素数. 求 $a+b+c$ 的最小值.\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 31 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n小明同学任选一个各项均为非零整数的等差数列 $\\left\\{a_{n}\\right\\}$ ，随后他列出 2020 个关于 $x$ 的一元二次方程:\n$$\nE_{i}: a_{i} x^{2}+a_{i+1} x+a_{i+2}=0(i=1,... | 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对正整数 $n$ 及实数 $x(0 \\leqslant x<n)$, 定义 \n$$ \nf(n, x)=(1-\\{x\\}) \\cdot C_{n}^{[x]}+\\{x\\} \\cdot C_{n}^{[x]+1}, \n$$ \n... | 74 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $A=\\{1,2,3\\}, B=\\{2 x+y \\mid x, y \\in A, x<y\\}, C=\\{2 x+y \\mid x, y \\in A, x>y\\}$ ，则 $B \\cap C$ 的所有元素之和为 $\\qquad... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若正整数 $a, b, c$ 满足 $2017 \\geqslant 10 a \\geqslant 100 b \\geqslant 1000 c$ ，则数组 $(a, b, c)$ 的个数为\n\nRemember to put your answ... | 574 |
[
{
"content": "Solve the following math problem step by step. 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 $\\left\\{a_{n}\\right\\}: a_{1}=7, \\frac{a_{n+1}}{a_{n}}=a_{n}+2, n=1,2,3, \\cdots$. 求满足 $a_{n}>4^{2018}$ 的最小正整数 $n$.\n... | 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设 $a_{i}(i=1,2,3,4)$ 均为实数, 若集合 $\\left\\{a_{1}, a_{2}, a_{3}, a_{4}\\right\\}$ 的所有非空真子集的元素之和为 28, 则 $a_{1}+a_{2}+a_{3}+a_{4}=$... | 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有长为 $2^{n}(n=0,1, \\cdots, 1009)$ 的线段各三条, 则由这 3030 条线段能构成不全等的三角形的个数为 $\\qquad$ (用数字作答).\n\nRemember to put your answer on its ... | 510555 |
[
{
"content": "Solve the following math problem step by step. 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 $\\odot O: x^{2}+y^{2}=5$ 与抛物线 $C$ : $y^{2}=2 p x(p>0)$ 交于点 $A\\left(x_{0}, 2\\right), A B$ 为 $\\odot O$ 的直径，过点 $B$ 的直线与抛物线... | 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已知函数 $f: \\mathbf{R} \\rightarrow \\mathbf{R}$, 若对任意实数 $x, y, z$, 都有\n$\\frac{1}{3} f(x y)+\\frac{1}{3} f(x z)-f(x) f(y z) \\g... | 1650 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $\\odot O_{1}: x^{2}+y^{2}=2$ 和 $\\odot O_{2}:(x-3)^{2}+$ $y^{2}=5$ 在第一象限内的公共点为 $A$, 过点 $A$ 的直线分别交 $\\odot O_{1} 、 \\odot O... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n由 $1,2, \\cdots, 9$ 这九个正整数构成的所有圆排列中, 任意相邻两数之积均不超过 60 的圆排列的个数为 $\\qquad$.\n\nRemember to put your answer on its own line after ... | 21600 |
[
{
"content": "Solve the following math problem step by step. 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 $n \\in \\mathbf{Z}_{+}$. 当 $n>100$ 时， $\\sqrt{n^{2}+3 n+1}$的小数部分的前两位数为 $\\qquad$.\n\nRemember to put your answer on its own... | 49 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $n$ 是小于 100 的正整数, 且满足 $\\frac{1}{3}\\left(n^{2}-1\\right)+\\frac{1}{5} n$ 为整数. 则符合条件的所有正整数 $n$ 的和为 $\\qquad$.\n\nRemember to... | 635 |
[
{
"content": "Solve the following math problem step by step. 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 $x+\\sin x \\cdot \\cos x-1=0$ ,$2 \\cos y-2 y+\\pi+4=0$ . 则 $\\sin (2 x-y)$ 的值为 $\\qquad$.\n\nRemember to put your answer o... | -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在矩形 $A B C D$ 中, $A B=3, A D=4, P$为矩形 $A B C D$ 所在平面上一点，满足 $P A=2$ ， $P C=\\sqrt{21}$. 则 $\\overrightarrow{P B} \\cdot \\overr... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThere are $2 n$ students in a school $(n \\in \\mathbb{N}, n \\geq 2)$. Each week $n$ students go on a trip. After several 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\n对任意正整数 $n$, 连结原点 $O$ 与点 $A_{n}(n, n+3)$, 用 $f(n)$ 表示线段 $O A_{n}$ 上的整点个数(不计端点), 试求 $f(1)+f(2)+\\cdots+f(1990)$.\n\nRemember to ... | 1326 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在坐标平面上, 横坐标和纵坐标均为整数的点称为整点, 对任意正整数 $n$, 连接原点 $O$ 与点 $A_{n}(n, n+3)$, 用 $f(n)$ 表示线段 $O A_{n}$ 上除端点外的整点个数,则 $f(1)+f(2)+\\cdots+f(... | 1326 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$1991^{2000}$ 除以 $10^{6}$ ，余数是 $\\qquad$.\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "use... | 880001 |
[
{
"content": "Solve the following math problem step by step. 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 $z_{1}, z_{2}$ 满 足 $\\left|z_{1}\\right|=\\left|z_{1}+z_{2}\\right|=3,\\left|z_{1}-z_{2}\\right|=3 \\sqrt{3}$ ，则$\\log _... | 4000 |
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