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过抛物线 $E: y^{2}=2 x$ 的焦点 $F$ 作两条斜率之积为 $-\\frac{1}{2}$ 的直线 $l_{1}, l_{2}$, 其中 $l_{1}$ 交 $E$ 于 $A, C$ 两点, $l_{2}$ 交 $E$ 于 $B, D$ ... | 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过抛物线 $E: y^{2}=2 x$ 的焦点 $F$ 作两条斜率之积为 $-\\frac{1}{2}$ 的直线 $l_{1} 、 l_{2}$, 其中, $l_{1}$ 与抛物线 $E$ 交于 $A 、 C$ 两点, $l_{2}$ 与抛物线 $E$... | 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一珍稀物种出现在地球，对每个珍稀生物，每天有如下事件发生：有 $p(0 \\leq p \\leq 1)$ 的概率消失，有 $\\frac{1-p}{3}$的概率保持不变，有 $\\frac{1-p}{3}$ 的概率分裂成两个，有 $\\frac{1-... | 22 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n有一人进行投篮训练，投篮 5 次，失误一次扣 1 分，进一次得 1 分，连进 2次得 3 分，连进 3 次得 5 分. 若投篮的命中率为 $\\frac{2}{5}$ ，则投篮 3 次恰好得 2 分的概率为 $\\qquad$. 原始的答案是\\fr... | 149 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n从正九边形中任取三个顶点构成三角形, 则正九边形的中心在三角形内的概率为 $\\qquad$. 原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put your answer o... | 19 |
[
{
"content": "Solve the following math problem step by step. 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, $\\sin ^{2} A+\\sin ^{2} C=2018 \\sin ^{2} B$, 则 $\\frac{(\\tan A+\\tan C) \\tan ^{2} B}{\\tan A+\\tan... | 2019 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若 $x, y$ 为实数, 则 $|2 x+y|,|x-y|,|1+y|$ 这三个数中的最大数的最小值是$\\qquad$.原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值\n\nRemember to put ... | 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\nGiven real numbers $b_0,b_1,\\ldots,b_{2019}$ with $b_{2019}\\neq0$, let $z_1,z_2,\\ldots,z_{2019}$ be the roots in the comple... | 4037 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nSuppose that $X_1,X_2,\\ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S=... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p$ be an odd prime number, and let ${\\Bbb {F}}_p$ denote the field of integers modulo $p$. Let ${\\Bbb {F}}_p[x]$ be the... | 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 smallest constant C such that for every real polynomial $P(x)$ of degree 3 that has a root in the interval $[0,1]$, $... | 11 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThe 30 edges of a regular icosahedron are distinguished by labeling them $1,2,…,30$. How many different ways are there to pain... | 35 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor a positive integer $N$, let $f_N^{1}$ be the function defined by\n$$f_N(x)=\\sum_{n=0}^N\\frac{N+1/2-n}{(N+1)(2n+1)}\\sin(... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to ... | 14 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that $$a^xb^{1-x}\\leq a\\frac... | 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 $Answer is the answer to the problem.\n\nSuppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane ... | 8 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor all $n\\geq1$, let $$a_n=\\sum_{k=1}^{n-1}\\frac{\\sin\\left(\\frac{(2k-1)\\pi}{2n}\\right)}{\\cos^2\\left(\\frac{(k-1)\\p... | 11 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet A be a $2n\\times2n$ matrix, with entries chosen independently at random. Every entry is chosen to be 0 or 1, each with pr... | 6 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $\\mathscr{F}$ be the set of functions $f(x,y)$ that are twice continuously differentiable for $x\\geq1,y\\geq1$ and that ... | 7 |
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nA line in the plane of a triangle $T$ is called an equalizer if it divides $T$ into two regions having equal area and equal pe... | 24 |
[
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"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor $0\\leq p\\leq1/2$,let $X_1,X_2,\\ldots$ be independent random variables such that $$X_i=\\begin{cases}1&with~ probabilit... | 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\nFor each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1,\\sqrt{2k}).$ Evaluate $$\\... | 19 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate \\(\\int_{0}^{a} \\int_{0}^{b} e^{\\max \\left(b^{2} x^{2}, a^{2} y^{2}\\right)} d y d x\\), where \\(a\\) and \\(b\\... | 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\nTwo real numbers x and y are chosen at random in the interval $(0,1)$ with respect to the uniform distribution. What is the pr... | 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有一个圆柱形水杯（厚度忽略不计），底面直径为 8, 高为 $8 \\sqrt{3}$. 水杯中有适量水,缓慢倾斜水杯至水恰好溢出，此时，水杯的母线与水平面的夹角为 $\\frac{\\pi}{3}$. 继续缓慢倾斜水杯, 停止倒水时水杯的母线与水平面的... | 134 |
[
{
"content": "Solve the following math problem step by step. 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|=\\left|z_{1}+2 z_{2}\\right|$ ，且 $\\overline{z_{1}} z_{2}=a(2-i)$ ( $a$ 为非零实数). ... | 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已知方程 $4 x|x|-a x+1=0$ 恰有三个解. 请给出$a$的最小整数值。\n\nRemember to put your answer on its own line after \"Answer:\".",
"role": "us... | 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某城镇有 $n(n \\geqslant 3)$ 座村庄,某些村庄之间有道路相连，且每条道路仅连接两座村庄，任两座村庄之间至多有一条路，同时，路交叉处没有十字路口，但有桥或隧道。已知任意两村庄之间可通过道路到达，但由于某项自行车赛事关闭任意环路之后，前... | 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如图 1, 在棱锥 $P-A B C D$ 中, 底面 $A B C D$ 为平行四边形， $P A \\perp$ 平面 $A B C D$ ， $\\angle A B C=\\theta\\left(\\theta \\in\\left(0, \... | 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$a_{1},a_{2},\text{...},a_{2021}$均为非负实数，满足$a_{1}+a_{2}+\text{...}+a_{2021}=1$。定义：$S=\text{...}\frac{a_{i}}{i},T=\text{...}\fr... | 2055 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知四边形 $A B C D$ 为圆柱轴截面，母线 $A B$ 长为 1, 底面直径 $B C$ 长为 $\\sqrt{2}$, $E$ 为底面圆周上的动点， $P$ 、 $Q$ 分别为点 $B$ 在 $A E 、 A C$ 上的射影。则当三棱锥 $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已知非负实数 $x$、$y$、$z$ 满足 $x+y+z=1$, 设 $S=\\frac{1-x}{1+x}+\\frac{1-y}{1+y}+\\frac{1-z}{1+z}$. 求 $S$ 的最小值和最大值。原始答案分别为\\dfrac{m_1}{... | 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在正三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, $A B=$ $B C=C A=A A_{1}=2$, 棱 $A_{1} C_{1}$ 上的点 $D$ 满足 $B C_{1} / /$平面 $A B_{1} \\dot{D}$ 。... | 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已知椭圆 $\\Gamma: \\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1, F$ 为 $\\Gamma$ 的右焦点。过 $F$ 作直线交 $\\Gamma$ 于 $M 、 N$ 两点，直线 $l$ 过点 $F$ 且与直线 $... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知点 $P$ 在椭圆 $\\frac{x^{2}}{2}+y^{2}=1$ 上, 记椭圆上顶点和右顶点分别为 $A$ 和 $B$. 则 $\\triangle A B P$ 面积的最大值的2倍减去$\\sqrt{2}$是 $\\qquad$。\n\n... | 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已知四面体 $A B C D$ 的外接球 $\\Gamma$ 的半径为 3, 球心为 $O$, 其中 $A B$ 为球 $\\Gamma$ 的直径, $C D=3, \\angle A O C=\\theta \\in\\left[\\frac{\\p... | 27 |
[
{
"content": "Solve the following math problem step by step. 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)=8\\sqrt{15}(7 \\sin x+\\sin 2 x)$ 的最大值为 $\\qquad$\n\nRemember to put your answer on its own line after \"Answer:\".",... | 225 |
[
{
"content": "Solve the following math problem step by step. 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)=\\frac{\\sin 2 x}{2+\\sin x+\\cos x}$ 的最小值减去1得到的结果的$\\sqrt{2}$倍\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已知点 $C \\in\\left\\{(x, y) \\mid x^{2}+y^{2}=1, y \\geq 0\\right\\}, A(-1,0),B(1,0)$, 延长 $A C$ 至 $D$ 使 $|C D|=3|B C|$, 那么点 $D... | 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解方程组 $\\left\\{\\begin{array}{l}x y=2 x+y-1 \\\\ y z=2 z+3 y-8. \\\\ z x=4 z+3 x-8\\end{array}\\right.$中x所有的可能值之和。\n\nRemember... | 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 $A, B$ 为抛物线 $y=1-x^{2}$ 上位于 $y$ 轴两侧的点, 求该抛物线过点 $A, B$ 的切线与 $x$ 轴围成的图形面积的最小值的$3\\sqrt{3}$倍。\n\nRemember to put your answer o... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n数列 $\\left\\{a_{n}\\right\\}$ 满足 \n$$ \na_{0}=1, a_{n+1}=\\sum_{\\substack{i, j, k \\geq 0 \\\\ i+j+k=n}} a_{i} a_{j} a_{k} ... | 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在正三棱雉 $P-A B C$ 中, 底面是边长为 1 的正三角形. 若点 $P$ 到平面 $A B C$ 的距离为 $x$, 异面直线 $A B$ 与 $C P$ 的距离为 $y$, 则 $\\lim _{x \\rightarrow+\\infty... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nEvaluate $29 \\frac{27}{28} \\times 27\\frac{14}{15}$. The original answer is in the form of $a + \\frac{m}{n}$, where a is th... | 1358 |
[
{
"content": "Solve the following math problem step by 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 a language college, $72\\%$ students can speak Chinese, $65\\%$ students can speak English, and $10\\%$ students can speak ... | 147 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nReplace the letters $a$, $b$, $c$ and $d$ in the following expression with the numbers $1$, $2$, $3$ and $4$, without repetiti... | 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\nFind all pairs of primes $(p,q)$ for which $p-q$ and $pq-q$ are both perfect squares. Please provide the value of $p + q + pq$... | 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\nFind all triples of positive integers $(x,y,z)$ that satisfy the equation \n\n$$\\begin{align*} 2(x+y+z+2xyz)^2=(2xy+2yz+2zx+... | 6 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nThree distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choic... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定正整数 $n$. 求最小的实数 $\\lambda$, 使得存在区间 $[0,1]$ 内的实数 $a_{1}, a_{2}$, $\\cdots, a_{n}$, 对于满足 $0 \\leqslant x_{1} \\leqslant x_{2} ... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine the smallest number $M$ such that the inequality\n$$\\begin{vmatrix}ab(a^2-b^2)+bc(b^2-c^2)+ca(c^2-a^2)\\end{vmatrix... | 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\nFind all functions $f\\colon(0,\\infty)\\to(0,\\infty)$ such that $\\frac{f(p)^2+f(q)^2}{f(r^2)+f(s^2)}=\\frac{p^2+q^2}{r^2+s^... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nOn some planet, there are $2^N$ countries ($N \\ge 4)$. Each country has a flag $N$ units wide and one unit high composed of $... | 17 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor any set $A = \\{a_1, a_2, a_3, a_4\\}$ of four distinct positive integers with sum $s_A = a_1 + a_2 + a_3 + a_4$, let $p_A... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy $f^{g(n)+1}(n)+g^{f(n)}(n)=f... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n \\geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself such that for all integers $... | 0 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor each positive integer $k$, let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which the... | 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\nFind all the functions $f: \\mathbb{Z} \\to \\mathbb{Z}$ such that $$f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)$$ for ... | 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\nIn a $999 × 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_1, C_2... | 3980000000000 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nCall admissible a set $A$ of integers that has the following property: If $x,y\\in A$ (possibly $x=y)$ then $x^2+kxy+y^2\\in A... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all triples $(x, y, z)$ of positive integers such that $x ≤ y ≤ z$ and $x^3(y^3 + z^3) = 2012(xyz + 2)$. Please provide t... | 505 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all $n\\in\\mathbb{N}$ for which there exist nonnegative integers $a_1,a_2,\\cdots,a_n$ such that $\\frac{1}{2^{a_{1}}}+\... | 37 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the least positive integer $n$ for which there exists a set $\\{s_1,s_2,\\ldots,s_n\\}$ consisting of $n$ distinct positi... | 39 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all functions $f:\\mathbb{Z}\\to\\mathbb{Z}$ satisfying $f\\left(f(m)+n\\right)+f(m)=f(n)+f(3m)+2014\\quad(1)$ for a... | 3007 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nDetermine all functions $f:\\mathbb{Z}\\to\\mathbb{Z}$ with the property that $$f\\big(x-f(y)\\big)=f\\big(f(x)\\big)-f(y)-1\\... | 2 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $n$ be a fixed positive integer. Find the maximum possible value of $$\\sum\\limits_{1\\leqslant r<s\\leqslant2n}(s-r-n)x_... | 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 $Answer is the answer to the problem.\n\n$x_{1}, x_{2}, x_{3}, x_{4}$ 为互不相等的正实数, $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 为 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意顺序排列, $X=\\max ... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n若一个 $k$ 项等差数列的公差是 4, 且首项的平方与除首项外的各项和小于 100 ，则 $k$ 的取值范围是哪些整数的集合？请提供该集合中所有元素的和。\n\nRemember to put your answer on its own line ... | 33 |
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{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $x^{2}=4 y, M(2,2)$ ，过 M 点的直线交抛物线于 $A, B$ 两点，过 $A, B$ 两点作抛物线的切线交与点 P ，求 $\triangle A B P$ 的最小面积。请给出最小面积的数值。\n\nRemember to ... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nGiven the functions $$ f(x)=\\frac{x}{\\ln x}, $$ and $g(x)=k(x-1)$ where $k \\in \\mathbf{R}$.\n(1) Prove that for any $k \\i... | 1 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n两个半径都是 1 的球 $O_{1}$ 和球 $O_{2}$ 相切，且均与直二面角 $\\alpha-l-\\beta$ 的两个半平面都相切，另有一个半径为 $r(r<1)$ 的小球 $O$与这个二面角的两个半平面也都相切, 同时,与球 $O_{1}$... | 10 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设动圆圆心在抛物线 $y=\\frac{1}{4} x^{2}$ 上, 半径等于该圆圆心的纵坐标. 求所有这样的圆上点的集合的边界条件。请给出边界条件中最大y值的答案。\n\nRemember to put your answer on its own... | 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 $a 、 b$ 都是正实数, $A=\\frac{a+b}{2}, B=\\frac{2}{\\frac{1}{a}+\\frac{1}{b}}$. 若 $A+B=a-b$, 则 $\\frac{a}{b}=$ 原始答案的形式为 $a+b\\sqr... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n现代社会对破译密码的要求越来越高。在密码学中，直接可以看到的内容为明码，对明码进行某种处理后得到的内容为密码。有一种密码将英文的 26 个字母 $a, b, \\cdots, z$ （不论大小写）依次对应 $1,2, \\cdots, 26$ 这 26... | 54 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $P$ 为直线 $y=x-2$ 上的动点，过 $P$ 作抛物线 $y=\\frac{1}{2} x^{2}$ 的切线，切点分别为 $A 、 B$.\n(1) 证明: 直线 $A B$ 过定点;\n(2) 求 $\\triangle P A B$ 面... | 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设不经过坐标原点 $O$ 的直线 $l$ 与圆 $x^{2}+y^{2}=1$ 交于不同的两点 $P 、 Q$. 若直线 $P Q$ 的斜率是直线 $O P$ 和 $O Q$ 斜率的等比中项,求 $\triangle P O Q$ 面积 $S$ 的取值... | 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数列 $\\left\\{a_{n}\\right\\}$ 满足: $a_{0}=\\sqrt{6}, a_{n+1}=\\left[a_{n}\\right]+\\frac{1}{\\left\\{a_{n}\\right\\}}$ (其中 $\\l... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知正实数 $x, y$ 满足: $\\frac{1}{x+3 y}+\\frac{1}{2 x+y}=1$, 则 $x+y$ 的最小值是。原始答案是 \\frac{m+2\\sqrt{n}}{k} 的格式，请给出 m + n + k 的值。\n\nR... | 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 $\\triangle A B C$ 的外接圆圆心为 $O$, 且 $3 \\overrightarrow{O A}+4 \\overrightarrow{O B}+5 \\overrightarrow{O C}=\\mathbf{0}$, 则 $... | 90 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n正四面体的 4 个表面上分别写有数字 $1,2,3,4$, 将 4 个这样的密度均匀的正四面体同时投掷于桌面上，与桌面接触的 4 个面上的 4 个数的和能被 4 整除的概率是 。请给出答案为\\(\\frac{m}{n}\\)形式时，m + n的值。\... | 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已知正实数 $x, y$ 满足: $\\frac{1}{x+3 y}+\\frac{1}{2 x+y}=1$, 则 $x+y$ 的最小值是 \\( \\frac{3+2\\sqrt{2}}{5} \\) 的形式。请给出 \\( m + n + k \\... | 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在三棱雉 $D-A B C$ 中, $\\angle B A D=30^{\\circ}$, $\\angle A C B=\\angle A B D=90^{\\circ}, C A=C B$. 若点 $C$ 在平面 $A B D$ 上的射影恰在 $... | 9 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知\n$$\n(1+\\sqrt{3})^{n}=a_{n}+b_{n} \\sqrt{3},\n$$\n其中, $a_{n}, b_{n}$ 为整数. 则 $\\lim _{n \\rightarrow+\\infty} \\frac{a_{n}}... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知函数 $f(x)=\\frac{x^{3}+3 x}{3 x^{2}+1}$, 数列 $\\left\\{x_{n}\\right\\}$ 满足 $x_{1}=2, x_{n+1}=f\\left(x_{n}\\right)\\left(n \\i... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n设 $x 、 y 、 z$ 为正实数. 求\n$$\n\\left(x+\\frac{1}{y}+\\sqrt{2}\\right)\\left(y+\\frac{1}{z}+\\sqrt{2}\\right)\\left(z+\\frac{1}{x}... | 34 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind all $r>0$ such that whenever $f: \\mathbb{R}^{2} \\rightarrow \\mathbb{R}$ is a differentiable function such that $|\\ope... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFind the number of positive integers $x$ satisfying the following two conditions: 1. $x<10^{2006}$; 2. $x^{2}-x$ is divisible ... | 12 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_{i}=i^{k}+i$ for $i=0,1, \\\\ldots, p... | 5 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nFor $n=1,2, \\ldots$ let\n$$\nS_{n}=-\\log \\left(\\sqrt[n^{2}]{1^{1} \\cdot 2^{2} \\cdot \\ldots \\cdot n^{n}}\\right)+\\log ... | 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\nLet for $j=0, \\ldots, n, a_{j}=a_{0}+j d$, where $a_{0}, d$ are fixed real numbers. Put\n$$\nA=\\left(\\begin{array}{ccccc}\n... | 8 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\nLet $A_{n}=\\{1,2, \\ldots, n\\}$, where $n \\geq 3$. Let $\\mathcal{F}$ be the family of all non-constant functions $f: A_{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已知函数 $f(x)=a \\sin x-\\frac{1}{2} \\cos 2 x+a-\\frac{3}{a}+\\frac{1}{2}, a \\in \\mathbf{R}, a \\neq 0$.\\n（1）若对任意 $x \\in \\m... | 37 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n已知 $x>0, y>0$, 且满足 $\\left\\{\\begin{array}{l}\\cos ^{2}(\\pi x)+2 \\sin (\\pi y)=1 \\\\ \\sin (\\pi x)+\\sin (\\pi y)=0 \\\\ ... | 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 $x_{1}, x_{2}, x_{3}$ 是非负实数，满足 $x_{1}+x_{2}+x_{3}=1$ ，求 $(x_{1}+3 x_{2}+5 x_{3})(x_{1}+\\frac{x_{2}}{3}+\\frac{x_{3}}{5})$ 的... | 15 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n长为 $\\sqrt{2}$, 宽为 1 的矩形, 以它的一条对角线所在的直线为轴旋转一周, 求得到的旋转体体积。原始答案为\\frac{m \\sqrt{p} \\pi}{n}的格式，请给出m + n+p的值。\n\nRemember to put ... | 98 |
[
{
"content": "Solve the following math problem step by step. 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 $\\log _{a} \\sqrt{2}<1$, 请给出$a$ 的取值范围。原始答案形式为$(a,b) \\cup(\\sqrt{c},+\\infty)$,计算a+b+c的值。\n\nRemember to put your answer on... | 3 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n给定整数 $n \\geq 2$. 设非负实数 $a_{1}, a_{2}, \\cdots, a_{n}$ 满足 \n$$ \na_{1} \\geq a_{2} \\geq \\cdots \\geq a_{n}, \\quad a_{1}+a... | 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等差数列 $\\left\\{a_{n}\\right\\}$ 满足 $a_{2021}=a_{20}+a_{21}=1$，则 $a_{1}$ 的值为原始答案的格式为 \\( \\frac{m}{n} \\)，请给出 m + n 的值。\n\nReme... | 5982 |
[
{
"content": "Solve the following math problem step by step. 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 $f(x)$ 满足: 对任意非零实数 $x$, 均有 $f(x)=f(1) \\cdot x+\\frac{f(2)}{x}-1$, 则 $f(x)$ 在 $(0,+\\infty)$ 上的最小值为 $\\qquad$.答案格式为$\\sqrt... | 4 |
[
{
"content": "Solve the following math problem step by step. The last line of your response should be of the form Answer: \\boxed{$Answer} where $Answer is the answer to the problem.\n\n在 $\\triangle A B C$ 中, $A B=1, A C=2, B-C=\\frac{2 \\pi}{3}$, 则 $\\triangle A B C$ 的面积为 $\\qquad$.答案格式为\\frac{a \\sqrt{b}}{c}... | 20 |
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