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设复数 $w=\\left(\\frac{a+\\mathrm{i}}{1+\\mathrm{i}}\\right)^{2}$，其中 $a$ 为实数，若 $w$ 的实部为 2，求 $w$ 的虚部。原始答案为-\\frac{k}{m}的格式，请给出k+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 $Answer is the answer to the problem.\n\n在 \\(\\triangle ABC\\) 中, 若三边长 \\(a, b, c\\) 满足 \\(a+c=3b\\), 则 \\(\\tan \\frac{A}{2} \\tan \\frac{C}{2}\\) 的值为多少？答案的形式为\\(\\f... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$A, B$ 是抛物线 $y=3-x^{2}$ 上关于直线 $x+y=0$ 对称的相异两点，求 $|A B|$ 的值。答案的形式为 $k \\sqrt{m}$，请给出 $k + m$ 的值。\n\nRemember to put your answer... | 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 $\\boldsymbol{a} 、 \\boldsymbol{b} 、 \\boldsymbol{c}$ 为同一平面内的三个单位向量，且 $a \\perp b$。求 $(\\boldsymbol{c}-\\boldsymbol{a}) \\cd... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知数列 $\\left\\{a_{n}\\right\\}$ 满足 $a_{1}=1, a_{n+1}=n+1+a_{n}\\left(n \\in \\mathbf{N}^{*}\\right)$. 若 $[x]$ 表示不超过实数 $x$ 的最大整... | 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 $Answer is the answer to the problem.\n\n若 \\(0<x<\\frac{\\pi}{2}\\), 且 \\(\\frac{\\sin ^{4} x}{9}+\\frac{\\cos ^{4} x}{4}=\\frac{1}{13}\\), 求 \\(\\tan x\\) 的值。答案的形式为\... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 \\(0<x<\\frac{\\pi}{2}\\), 且 \\(\\frac{\\sin ^{4} x}{9}+\\frac{\\cos ^{4} x}{4}=\\frac{1}{13}\\)，求 \\(\\tan x\\) 的值。答案的形式为\\... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 \\(x 、 y \\in \\mathbf{R}\\), 且\n\\(\\log _{4}(x+2 y)+\\log _{4}(x-2 y)=1\\).\n求 \\(x-|y|\\) 的最小值。答案形式为\\(\\sqrt{k}\\)，请给出k的... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在边长为 8 的正方形 $A B C D$ 中, $M$ 为边 $B C$ 的中点, $N$ 为边 $D A$ 上一点, 且 $D N=3 N A$. 若对于常数 $m$ ，在正方形 $A B C D$ 的边上恰有六个不同的点 $P$ ，使得 $\\o... | 7 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 $f(x)=|2 x-4|-3 x+8$ $(x \\in \\mathbf{R})$ 。求 $f(x)$ 的反函数 $f^{-1}(x)$ 的解析式。答案的形式为\\begin{cases}-kx+m, x\\leqslant2;\\\\-... | 28 |
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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 $Answer is the answer to the problem.\n\n已知二面角 \\(\\alpha-l-\\beta\\) 的大小为 \\(30^{\\circ}\\)。求由平面 \\(\\alpha\\) 上的圆在平面 \\(\\beta\\) 上的正射影得到的椭圆的离心率。原始答案的形式为\\(\\frac{k}... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 \\(f(x)=x^{2}-2 t x+t\\)。当 \\(x \\in [-1,1]\\) 时，记 \\(f(x)\\) 的最小值为 \\(m\\)。求 \\(m\\) 的最大值。答案的形式为\\(\\frac{k}{m}\\)，请给出k+... | 5 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设方程\n$$\n1+x-\\frac{x^{2}}{2}+\\frac{x^{3}}{3}-\\frac{x^{4}}{4}+\\cdots-\\frac{x^{2018}}{2018}=0\n$$\n的所有实数根均在区间 $[a, b](a 、b ... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 \\( f(x) \\) 是定义在实数集上的周期为 2 的函数，且是偶函数。已知当 \\( x \\in [2,3] \\) 时，\\( f(x) = x \\)。请根据已知条件，推导出当 \\( x \\in [-2,0] \\) 时，\\( f... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 \\( f(x) \\) 是定义在实数集上的周期为 2 的函数，且是偶函数。已知当 \\( x \\in [2,3] \\) 时, \\( f(x)=x \\)。求当 \\( x \\in [-2,0] \\) 时, \\( f(x) \\) 的解... | 4 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $a$ 是正整数， $a<100$ ，并且 $a^{3}+23$ 能被 24 整除。求满足条件的 $a$ 的个数。\n\nRemember to put your answer on its own line after \"Answer:\"."... | 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设复平面上单位圆内接正 20 边形的 20 个顶点所对应的复数依次为 $Z_{1}, Z_{2}, \\cdots, Z_{20}$, 求复数 $Z_1^{1995}, Z_2^{1995}, \\cdots, Z_{20}^{1995}$ 所对应的不... | 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已知方程 $|x-2 n|=k \\sqrt{x}(n \\in \\mathbf{N})$ 在区间 $(2 n-1,2 n+1]$ 上有两个不相等的实根，求 $k$ 的取值范围。答案的形式为 $0<k \\leqslant \\frac{1}{\\s... | 3 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知两个实数集合 $A=\\left\\{a_{1}, a_{2}, \\cdots, a_{100}\\right\\}$ 与 $B=\\left\\{b_{1}, b_{2}, \\cdots, b_{50}\\right\\}$, 若从 $A$ ... | 148 |
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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删去正整数数列 $1,2,3, \\cdots \\cdots$ 中的所有完全平方数, 得到一个新数列。求这个数列的第 2003 项。\n\nRemember to put your answer on its own line after \"Ans... | 2048 |
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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 obvious approximations to the length of the perimeter of the ellipse with semi-axes $a$ and $b$ are $\\pi(a+b)$ and $2 \\p... | 2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n过抛物线 \\(y^{2}=8(x+2)\\) 的焦点 \\(F\\) 作倾斜角为 \\(60^{\\circ}\\) 的直线，若此直线与抛物线交于 \\(A 、 B\\) 两点，弦 \\(AB\\) 的中垂线与 \\(x\\) 轴交于点 \\(P\\... | 19 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 \\(x, y\\) 都在区间 \\((-2,2)\\) 内，且 \\(x y=-1\\)，求函数 \\(u=\\frac{4}{4-x^{2}}+\\frac{9}{9-y^{2}}\\) 的最小值。答案的形式为\\(\\frac{k}{m}\... | 17 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在四面体 \\(A B C D\\) 中，设 \\(A B=1\\)，\\(C D=\\sqrt{3}\\)，直线 \\(A B\\) 与 \\(C D\\) 的距离为 2，夹角为 \\(\\frac{\\pi}{3}\\)。求四面体 \\(A B C... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设等差数列的首项及公差均为非负整数，项数不少于 3 ，且各项的和为 $97^{2}$ ，求这样的数列共有多少个。\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 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已知数列 $\\left\\{x_{n}\\right\\}$ 满足 $x_{n+1}=x_{n}-x_{n-1}(n \\geqslant 2), x_{1}=a, x_{2}=b$, 记 $S_{n}=$ $x_{1}+x_{2}+\\cdots+... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设等差数列的首项及公差均为非负整数，项数不少于 3 ，且各项的和为 $97^{2}$ 。求这样的数列的个数。\n\nRemember to put your answer on its own line after \"Answer:\".",
... | 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 $E, F, G$ 分别是正四面体 $A B C D$ 的棱 $A B, B C, C D$ 的中点，求二面角 $C-F G-E$ 的大小。如果答案为 $\\pi-\\operatorname{arccot} x$，其原始答案的形式为\\frac{... | 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 $E 、 F 、 G$ 分别是正四面体 $A B C D$ 的棱 $A B 、 B C 、 C D$ 的中点。求二面角 $C-F G-E$ 的大小，并用表达式 $\\pi-\\operatorname{arccot} \\frac{\\sqrt{m... | 4 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 \\(0<x<\\frac{\\pi}{2}\\)，且 \\(\\frac{\\sin ^{4} x}{9}+\\frac{\\cos ^{4} x}{4}=\\frac{1}{13}\\)，求 \\(\\tan x\\) 的值。答案形式为 \\(... | 5 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对于实数 $t$, 已知等比数列 $\\left\\{a_n\\right\\}$ 的前三项依次为 $2 t, 5 t-1,6 t+2$, 且该数列的前 $n$ 项和为 $S_n$. 求满足不等式 $|S_n-\\frac{1}{65}|<1$ 的最大... | 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 $Answer is the answer to the problem.\n\n已知 \\(0<k<4\\)，直线 \\(l_1: k x-2 y-2 k+8=0\\) 和直线 \\(l_2: 2 x+k^2 y-4 k^2-4=0\\) 与两坐标轴围成一个四边形。求使这个四边形面积最小的 \\(k\\) 值。答案的形式为\\(\... | 9 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n对于实数 $t$, 已知等比数列 $\\left\\{a_{n}\\right\\}$ 的前三项依次为 $2 t, 5 t-1,6 t+2$, 且该数列的前 $n$ 项和为 $S_{n}$. 求满足不等式 $\\left|S_{n}-\\frac{1}... | 2 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n长度分别为 $1, a, a, a, a, a$ 的线段能成为同一个四面体的 6 条棱的充要条件是 $a$ 满足的条件是 \\frac{\\sqrt{k}}{m}，请给出k+m的值。\n\nRemember to put your answer on ... | 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 $Answer is the answer to the problem.\n\n记 \\(F(x, y) = (x-y)^2 + \\left( \\frac{x}{3} + \\frac{3}{y} \\right)^2 \\quad (y \\neq 0)\\)。求 \\(F(x, y)\\) 的最小值。原始答案的形式为\\(... | 23 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn the same side of a straight line, three circles are drawn as follows: a circle with a radius of $4$ inches is tangent to th... | 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 $Answer is the answer to the problem.\n\nBrianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her ... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n求$(x-1)(x-2)(x-3)(x-4)$的最小值，其中$x$为实数。请给出最小值的整数部分。\n\nRemember to put your answer on its own line after \"Answer:\".",
"rol... | -1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEquilateral triangle \\(ABC\\) has been creased and folded so that vertex \\(A\\) now rests at \\(A'\\) on \\(\\overline{BC}\\... | 48 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven a right triangle $ABC$ with legs $BC=3$ and $AC=4$, find the length of the shorter angle trisector from $C$ to the hypot... | 72 |
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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 $Answer is the answer to the problem.\n\nWhat is the maximum value of the function \\(\\frac{1}{|x+1| + |x+2| + |x-3|}?\\) The original answer is in \\(\\frac{k}{m}\\)... | 6 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA palindrome, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x + 32$ ar... | 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\nWhich value of $x$ satisfies the equation $\\sqrt[3]{6+\\sqrt{x}} + \\sqrt[3]{6-\\sqrt{x}} = \\sqrt[3]{3}$?\n\nRemember to put... | 45 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nConsider two concentric circles with radii \\(17\\) and \\(19\\). The larger circle has a chord, half of which lies inside the... | 14 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree digits are selected at random and without replacement from the set of digits $0$ through $9$. Find the probability that ... | 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 $Answer is the answer to the problem.\n\nIf \\(b > 1\\), \\(x > 0\\) and \\((2x)^{\\log_b 2} - (3x)^{\\log_b 3} = 0\\), find the value of \\(x\\). The original answer ... | 7 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nHow many real pairs $(x, y)$ are there such that\n\\[\nx^2 + 2y = 2xy \\\\\nx^3 + x^2y = y^2\n\\]\nFind the number of real pai... | 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\nHow many triples of positive integers $(a,b,c)$ are there such that $a!+b^3 = 18+c^3$?\n\nRemember to put your answer on its o... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the total number of solutions of the equation $xyz=2520$.\n\nRemember to put your answer on its own line after \"Answer:\... | 540 |
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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 $ \\log 2 = 0.3010$ and $ \\log 3 = 0.4771$, find the value of $ x$ when $ 3^{x+3} = 135$. If the answer is $y$, what is th... | 147 |
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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 $Answer is the answer to the problem.\n\nWhat is the sum \\( \\frac{1}{1 \\cdot 2 \\cdot 3} + \\frac{1}{2 \\cdot 3 \\cdot 4} + \\cdots + \\frac{1}{1996 \\cdot 1997 \\c... | 4003 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a > b > 0$... | 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\nLet $a$, $b$, $c$, $d$, and $e$ be distinct integers such that\n\\[ (6 - a)(6 - b)(6 - c)(6 - d)(6 - e) = 45. \\]\nFind the va... | 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 $Answer is the answer to the problem.\n\nAndrea inscribed a circle inside a regular pentagon and circumscribed a circle around it, calculating the area of the region b... | 1 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nTwo integers have a sum of $26$. When two more integers are added to the first two integers, the sum is $41$. Finally, when tw... | 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 $Answer is the answer to the problem.\n\nA circular grass plot with a diameter of 12 feet is intersected by a straight gravel path that is 3 feet wide. One edge of the... | 42 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$(a_n)_{n = 0}^\\infty$ is a sequence of integers. For every $n \\ge 0$, $a_{n + 1} = a_n^3 + a_n^2$. Find the maximum number ... | 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 $Answer is the answer to the problem.\n\nThe stronger Goldbach conjecture states that any even integer greater than 7 can be written as the sum of two different prime ... | 100 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; the... | 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\nAt each step, we change the positions of exactly two numbers from the sequence $1, 2, 3, 4, 5, 6, 7$. How many different arran... | 176 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each positive integer $n$, let $S(n)$ denote the sum of the digits of $n$. Determine how many values of $n$ satisfy the eq... | 4 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nWhat is the maximum area of an isosceles trapezoid that has legs of length \\(1\\) and one base twice as long as the other? Th... | 5 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nIn $\\triangle ABC$, $AB = 8$, $BC = 7$, $CA = 6$, and side $BC$ is extended to a point $P$ such that $\\triangle PAB$ is simi... | 9 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $S$ be the set of all those 2007-place decimal integers $\\overline{2s_1a_2a_3 \\ldots a_{2006}}$ which contain an odd num... | 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 $Answer is the answer to the problem.\n\nIn the equation \\(\\frac{x(x - 1) - (m + 1)}{(x - 1)(m - 1)} = \\frac{x}{m}\\), find the value of \\(m\\) such that the roots... | 3 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA particle of mass $m$ moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional ... | 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\nA vertical mass-spring oscillator is displaced $2.0\\, \\text{cm}$ from equilibrium. The $100\\, \\text{g}$ mass passes throug... | 140 |
[
{
"content": "Solve the following math problem step by 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 one can find a student with at least $k$ friends in any class which has 21 students such that at least two of any three of ... | 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\nIf $ABCD$ is a $2 \\times 2$ square, $E$ is the midpoint of $\\overline{AB}$, $F$ is the midpoint of $\\overline{BC}$, $\\over... | 22 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设\\(\\boldsymbol{a}、\\boldsymbol{b}、\\boldsymbol{c}\\)为同一平面内的三个单位向量，且 \\(\\boldsymbol{a}\\perp\\boldsymbol{b}\\)。求\\((\\boldsy... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n如图 1, 设 $P$ 为 $\\triangle A B C$ 内一点, 且 $\\boldsymbol{A P}$ $=\\frac{2}{5} \\boldsymbol{A B}+\\frac{1}{5} \\boldsymbol{A} \\bo... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n如图, 在棱长为 1 的正四面体 $A B C D$ 中, $G$ 为 $\\triangle B C D$ 的重心, $M$ 是线段 $A G$ 的中点, 求三棱锥 $M-B C D$ 的外接球的表面积除以 $\\pi$ 的值。答案的形式为\\(\\... | 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 $Answer is the answer to the problem.\n\n设 \\(x, y\\in\\mathbf{R}\\), 且\\(\\log _{4}(x+2 y)+\\log _{4}(x-2y)=1\\), 求\\(x-|y|\\)的最小值。原始答案是以\\(\\sqrt{k}\\)的形式给出，请给出k的值。\... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设函数$f(x)=x^{3}+a x^{2}+b x+c$（$a, b, c$均为非零整数），若 $f(a)=a^{3}, f(b)=b^{3}$，则 $c$ 的值是多少？\n\nRemember to put your answer on its o... | 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在凸四边形 $A B C D$ 中, 若 $B C=4, \\angle A D C=60^{\\circ}, \\angle B A D=90^{\\circ}$, 四边形 $A B C D$ 的面积等于 $\\frac{A B \\cdot C D... | 69 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n四个半径为 1 的球两两相切，则它们外切正四面体的棱长为多少？答案的形式为 $k(1+\\sqrt{m})$，请给出 $k+m$ 的值。\n\nRemember to put your answer on its own line after \"An... | 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已知 $k \\neq 1$ ，则等比数列 $a+\\log _{2} k, a+\\log _{4} k, a+\\log _{8} k$ 的公比是多少？答案的形式为\\frac{k}{m}，请给出k+m的值。\n\nRemember to put ... | 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 $Answer is the answer to the problem.\n\n$f(x)$ 是定义在实数集 $\\mathbf{R}$ 上的函数，满足 $\\forall x \\in \\mathbf{R}, 2 f(x)+f\\left(x^{2}-1\\right)=1$ ，求 $f(-\\sqrt{2})$ 的值。答案的... | 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已知正方形 $A B C D$ 的边长为 1, 若 $P_{1}, P_{2}, P_{3}, P_{4}$ 是正方形内部的 4 个点，使得 $\\triangle A B P_{1}, \\triangle B C P_{2}, \\triangle... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n使得 $p^{3}+7 p^{2}$ 为完全平方数的不大于 100 的素数 $p$ 的个数为多少？\n\nRemember to put your answer on its own line after \"Answer:\".",
"rol... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n函数 $f(x)=x(x+1)(x+2)(x+3)$ 的最小值是多少？\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
}... | -1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n不等式组 \\(\\left\\{\\begin{array}{l}y \\geqslant 2|x|-1 \\\\ y \\leqslant-3|x|+5\\end{array}\\right.\\) 所表示的平面区域的面积是多少？答案应为\\(\\... | 41 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $O$ 是凸四边形 $A B C D$ 对角线 $A C$ 和 $B D$ 的交点, $\\triangle A O B, \\triangle B O C$, $\\triangle C O D, \\triangle D O A$ 的周长相等... | 11 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若某三角形的三边长为三个连续正整数，且该三角形有一个角是另一个角的 2 倍，请找出这个三角形的三边长。答案的格式为(k, m, n)，请给出k + m + n的值。\n\nRemember to put your answer on its own l... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若函数 $f(t)=t^{2}+2t$ ，则点集 $\\left\\{(x, y) \\left\\lvert\\,\\left\\{\\begin{array}{l}f(x)+f(y) \\leqslant 2 \\\\ f(x) \\geqslan... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若数列 \\(\\left\\{a_{n}\\right\\}\\) 的前 \\(n\\) 项和 \\(S_{n}=n^{2} a_{n}\\)，且 \\(a_{1}=2019\\)，求 \\(a_{2018}\\) 的值。原始答案的形式为\\(\\f... | 1010 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若正数 \\(a, b\\) 满足 \\(a+b=1\\)，则求 \\(\\frac{1}{a}+\\frac{27}{b^{3}}\\) 的最小值。答案的形式为\\(\\frac{k+m \\sqrt{3}}{n}\\)，请计算k+m+n的值。\n\... | 62 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设实数 $x ,y$ 满足 $\\frac{x^{2}}{5}+\\frac{y^{2}}{4}=1$, 求 $\\sqrt{x^{2}+y^{2}-2 y+1}+\\sqrt{x^{2}+y^{2}-2 x+1}$ 的最小值。答案的形式为 $k\\s... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n15 人围坐在圆桌旁，选出 4 人使得其中任意两人都不相邻的选法数为多少？\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "user"
... | 450 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设集合 $S$ 中有 10 个元素, 从 $S$ 中每次随机选取 1 个元素, 取出后还放回 $S$ 中, 则取5次后出现重复元素的概率是（保留两位有效数字）。如果答案是 $x$，那么请给出 $100x$ 的值。\n\nRemember to put ... | 70 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n一个正整数 $n$ 称为具有 3 -因数积性质。若 $n$ 的所有正因数的乘积等于 $n^{3}$ 。求不超过 400 的正整数中具有 3 -因数积性质的数的个数。\n\nRemember to put your answer on its own l... | 51 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 \\(A\\) 是不超过 2018 的正整数组成的集合，对于正整数 \\(k\\)，用 \\(a_{k}\\) 表示所有可能的 \\(A\\)中 \\(k\\) 个数乘积的倒数之和。求 \\(a_{2}+a_{4}+\\cdots+a_{2018}... | 2019 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n用 $[x]$ 表示不超过实数 $x$ 的最大整数。例如 $[\\pi]=3$, $[-\\pi]=-4$ 。设 $n$ 为正整数。用 $a_{n}$ 表示当 $x \\in[0, n)$ 时, 函数 $f(x)=[x[x]]$ 的值域中的元素的个数。... | 63 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 \\(\\triangle A B C\\) 的面积为 1, \\(D, E\\) 分别为边 \\(BC, CA\\) 上的点, 且 \\(B D=\\frac{1}{3} B C\\), \\(C E=\\frac{1}{3} C A\\), ... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n$1 \\times 1!+2 \\times 2!+\\cdots+672 \\times 672!$ 被 2019 除的余数是多少？\n\nRemember to put your answer on its own line after \"An... | 2018 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若存在正整数 $k<n\\left(n \\geqslant 3, n \\in \\mathrm{~N}^{*}\\right)$, 使得 $\\mathrm{C}_{n}^{k-1}, \\mathrm{C}_{n}^{k}, \\mathrm{C... | 42 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在（2019 $\\cdot$ 2020）2021 的全体正因数中选出若干个，使得其中任意两个之积都不是完全平方数，则最多可选的正因数个数为多少？\n\nRemember to put your answer on its own line after... | 32 |
[
{
"content": "Solve the following math problem step by step. 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 $a_{1}=1, a_{2}=4$, 且 $a_{n}^{2}-$ $a_{n-1} a_{n+1}=2^{n-1}\\left(n \\geqslant 2, n \\in \\m... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 $a, b, c, d$ 是方程 $x^{4}+2 x^{3}+3 x^{2}+4 x+5= 0$ 的全部复数根, 求 $\\frac{a-1}{a+2}+\\frac{b-1}{b+2}+\\frac{c-1}{c+2}+\\frac{d-1}{... | -1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知等边 \\(\\triangle ABC\\) 的边长为1，过点 \\(C\\) 作以边 \\(AB\\) 为直径的圆的切线交线段 \\(AB\\) 的延长线于点 \\(D\\)，且 \\(AD > BD\\)，求 \\(\\triangle BC... | 21 |
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