ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Define a positive integer $n$ to be squarish if either $n$ is itself a perfect square or the distance from $n$ to the nearest perfect square is a perfect square. For example, 2016 is squarish, because the nearest perfect square to 2016 is $45^2=2025$ and 2025-2016=9 is a perfect square. (Of the positive integers between l and 10, only 6 and 7 are not squarish.) For a positive integer $N$, let $S(N)$ be the number of squarish integers between l and $N$, inclusive. Find positive constants $\alpha$ and $\beta$ such that $$\lim_{N\to\infty}\frac{S(N)}{N^{\alpha}}=\beta,$$ or show that no such constants exist. Please provide the sum of $12\alpha v$.
Answer: 25
Problem: For all $n\geq1$, let $$a_n=\sum_{k=1}^{n-1}\frac{\sin\left(\frac{(2k-1)\pi}{2n}\right)}{\cos^2\left(\frac{(k-1)\pi}{2n}\right)\cos^2\left(\frac{k\pi}{2n}\right)}.$$ Determine $$\lim_{n\to\infty}\frac{a_n}{n^3}.$$ The original answer is in the format $\frac{m}{\pi^{n}}$. Please provide the value of m + n.
Answer: 11
Problem: Determine the maximum value of the sum
$$S=\sum_{n=1}^{\infty}\frac{n}{2^{n}}\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}$$
over all sequences $a_{1},a_{2},a_{3},\cdots$ of nonnegative real numbers satisfying
$$\sum_{k=1}^{\infty}a_{k}=1.$$ The original answer is in \frac{m}{n} format, please give the value of m + n.
Answer: 5
Problem: Let $\times$ represent the cross product in $\mathbb{R}^3.$ For what positive integers $n$ does there exist a set $S\subset\mathbb{R}^3$ with exactly $n$ elements such that
$$S=\{v\times w:v,w\in S\}?$$ Please provide the sum of all possible values of $n$.
Answer: 8
Problem: Let $x_0=1$, and let $\delta$ be some constant satisfying $0<\delta<1.$ Iteratively, for $n=0,1,2,\ldots$, a point $x_{n+1}$ is chosen uniformly from the interval $[0,x_n].$ Let Z be the smallest value of $n$ for which $x_n<\delta.$ Find the expected value of Z, as a function of $\delta$. Please provide the value when $\delta = 10$. The answer should be an integer.
Answer: 0
Problem: Let A be a $2n\times2n$ matrix, with entries chosen independently at random. Every entry is chosen to be 0 or 1, each with probability 1/2. Find the expected value of $\det(A-A^{t})$ (as a function of n), where $A^{t}$ is the transpose of $A.$ The original answer is in \frac{(an)!}{b^{n}n!} format, please give the value of a+b.
Answer: 6
Problem: Evaluate the sum
$\sum_{k=0}^{\infty}\left(3\cdot\frac{\ln(4k+2)}{4k+2}-\frac{\ln(4k+3)}{4k+3}-\frac{\ln(4k+4)}{4k+4}-\frac{\ln(4k+5)}{4k+5}\right)$
$=3\cdot\frac{\ln2}{2}-\frac{\ln3}{3}-\frac{\ln4}{4}-\frac{\ln5}{5}+3\cdot\frac{\ln6}{6}-\frac{\ln7}{7}-\frac{\ln8}{8}-\frac{\ln9}{9}+3\cdot\frac{\ln10}{10}-\cdots.$
(As usual, $lnx$ denotes the natural logarithm of x.)
 The original answer is in $(\log m)^{n}$format, please give the value of m + n.
Answer: 4
Problem: Let $\mathscr{F}$ be the set of functions $f(x,y)$ that are twice continuously differentiable for $x\geq1,y\geq1$ and that satisfy the following two equations (where subscripts denote partial derivatives): $$\begin{array}{c}xf_x+yf_y=xy\ln(xy),\\x^2f_{xx}+y^2f_{yy}=xy.\end{array}$$ For each $f\in\mathscr{F}$,let $$m(f)=\min_{s\geq1}\left(f(s+1,s+1)-f(s+1,s)-f(s,s+1)+f(s,s)\right).$$ Determine $m(f)$, and show that it is independent of the choice of $f.$ The original answer is in the format of $m \ln n - \frac{a}{b}$, please give the value of m + n +a +b.
Answer: 7
Problem: Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$ for which $s_{0}=s_{2n}=0$ and $|s_j-s_{j-1}|=1$ for $j=1,2,\cdots,2n.$ Define $$q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_{j}},$$ and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over all $\mathbf{v}\in V_n.$ Evaluate $M(2020). The original answer is in \frac{m}{n} format, please give the value of m + n.
Answer: 4041
Problem: A line in the plane of a triangle $T$ is called an equalizer if it divides $T$ into two regions having equal area and equal perimeter. Find positive integers $a > b > c$, with $a$ as small as possible, such that there exists a triangle with side lengths $a, b, c$ that has exactly two distinct equalizers. Please provide the value of $a + b + c$.
Answer: 24
Problem: Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\leq y\leq x^{3}$,then$(f(x))^2\leq f(y)\leq(f(x))^3.$ Please provide the smallest value of c when assuming $f(x)=x^c$ and c is a specific constant value such that the function satisfies the given property.
Answer: 1
Problem: For $0\leq p\leq1/2$,let $X_1,X_2,\ldots$ be independent random variables such that  $$X_i=\begin{cases}1&with~ probability~p,\\-1&{with ~probability}~p,\\0&{with ~probability}~1-2p,\end{cases}$$  for all $i\geq1.$ Given a positive integer $n$ and integers $b,a_1,\ldots,a_n$, let $P(b,a_1,\ldots,a_n)$ denote the probability that $a_1X_1+\cdots+a_nX_n=b.$ For which values of $p$ is it the case that  $$P(0,a_1,\ldots,a_n)\geq P(b,a_1,\ldots,a_n)$$  for all positive integers $n$ and all integers $b,a_1,\ldots,a_n?$ The original answer is in the format $p \leq m / n$, please provide the value of m+n.
Answer: 5
Problem: Determine which positive integers n have the following property: For all integers m that are relatively prime to n, there exists a permutation $\pi:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ such that $\pi(\pi(k))\equiv mk(\bmod n)$ for all $k\in\{1,2,\ldots,n\}$. Provide the sum of all such positive integers n.
Answer: 3
Problem: For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1,\sqrt{2k}).$ Evaluate $$\sum_{k=1}^{\infty}(-1)^{k-1}\frac{A(k)}{k}.$$ The original answer is in the format $\frac{m\pi^j}{n}$, please provide the value of m + n + j.
Answer: 19
Problem: Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2,\ldots,n\},$ let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai+bj=n.$ Let S be the n-by-n matrix whose $(i,j)$ entry is $s(i,j).$ For example, when $n=5$, we have $S=\begin{bmatrix}6&3&2&2&2\\3&0&1&0&1\\2&1&0&0&1\\2&0&0&0&1\\2&1&1&1&2\end{bmatrix}.$ Compute the determinant of $S$ for $n=5$. The original answer is in the format $(-1)^{\lceil n / 2\rceil-1} 2\left\lceil\frac{n}{2}\right\rceil$, please give the value of the determinant for $n=2$.
Answer: 2
Problem: A $2 \times 3$ rectangle has vertices at $(0,0),(2,0),(0,3)$, and $(2,3)$. It rotates $90^{\circ}$ clockwise about the point $(2,0)$. It then rotates $90^{\circ}$ clockwise about the point $(5,0)$, then $90^{\circ}$ clockwise about the point $(7,0)$, and finally, $90^{\circ}$ clockwise about the point $(10,0)$. (The side originally on the $x$-axis is now back on the $x$-axis.) Find the area of the region above the $x$-axis and below the curve traced out by the point whose initial position is $(1,1)$. The original answer is in the form of \(\frac{m\pi}{n} + k\), please provide the value of \(m + n + k\).
Answer: 15
Problem: Evaluate \(\int_{0}^{a} \int_{0}^{b} e^{\max \left(b^{2} x^{2}, a^{2} y^{2}\right)} d y d x\), where \(a\) and \(b\) are positive. If \(a = 1\) and \(b = 1\), find the value of \(m + n\) in the expression \(\frac{e^{a^mb^n}-j}{ab}\), where the answer is in the format \(m + n + j\).
Answer: 5
Problem: A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. The original answer is in the form $\frac{a\sqrt{b}-c}{d}$, please give the value of a + b + c + d.
Answer: 14
Problem: Let f be a function on $[0, \infty)$, differentiable and satisfying  
$$  
f^{\prime}(x)=-3 f(x)+6 f(2 x)  
$$  
for $x>0$. Assume that $|f(x)| \leqslant e^{-\sqrt{x}}$ for $x \geqslant 0$ (so that $f(x)$ tends rapidly to 0 as x increases). For $n$ a non-negative integer, define  
$$  
\mu_{n}=\int_{0}^{\infty} x^{n} f(x) d x  
$$  
(sometimes called the $n$th moment of f ).  

Please express $\mu_{n}$ in terms of $\mu_{0}$ and give the value of $\mu_{2}$ when $\mu_{0} = 45$. Directly provide the integer value of $\mu_{2}$.
Answer: 32
Problem: For any pair $(x, y)$ of real numbers, a sequence $\left(a_{n}(x, y)\right)_{n \geq 0}$ is defined as follows:  $$\begin{aligned} a_{0}(x, y) & =x \\ a_{n+1}(x, y) & =\frac{\left(a_{n}(x, y)\right)^{2}+y^{2}}{2}, \quad \text { for all } n \geq 0. \end{aligned}$$  Find the area of the region $\left\{(x, y) \mid\left(a_{n}(x, y)\right)_{n \geq 0}\right.$ converges $\}$. The original answer is in the format $m+n\pi$, please provide the value of m+n.
Answer: 5
Problem: Two real numbers x and y are chosen at random in the interval $(0,1)$ with respect to the uniform distribution. What is the probability that the closest integer to $x / y$ is even? The original answer is in the form $(m-\pi)/n$, please provide the value of m+n.
Answer: 9
Problem: 设双曲线 $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1, A_{1} 、 A_{2}$ 为双曲线 $C$ 的左、右顶点, $F$ 为 $C$ 的焦点. 以坐标原点 $O$ 为圆心、 $|O F|$ 为半径的圆与 $C$ 的渐近线的一个交点为 $B$. 若 $\angle A_{1} B A_{2}=60^{\circ}$, 则 $C$ 的离心率的原始答案为$\frac{\sqrt{m}}{n}$，请给出m + n的值。
Answer: 24
Problem: 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夹角为 $\frac{\pi}{6}$. 则所倒出的水的体积的原始答案是 $\frac{m \sqrt{k} \pi}{n}$ 形式，请给出 m + n+k 的值。
Answer: 134
Problem: 已知数列 $\left\{(-1)^{n} \frac{3 n+2}{n(n+1) \cdot 2^{n+1}}\right\}$ 的前 $n$ 项和为 $S_{n}$. 计算 $S_{18}$ 的值。标准答案格式为$\frac{a}{b^{c} \times d}-\frac{e}{f}$，计算a+b+c+d+e+f的值。
Answer: 44
Problem: 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). 则 $\frac{z_{2}}{z_{1}}=\_$。已知原始答案为 $-\frac{a}{b}+\frac{c}{d} i$的形式，请给出 a+b+c+d 的值。
Answer: 16
Problem: 已知椭圆 $\frac{x^{2}}{5}+\frac{y^{2}}{4}=1$ 的右焦点为 $F$,上顶点为 $A, P$ 为椭圆上位于第一象限内的动点. 则 $\triangle A P F$ 面积的最大值为$\qquad$。答案的标准格式为$\sqrt{a}-b$，则a+b的值为多少？
Answer: 7
Problem: 设正四面体 $A B C D$ 的棱长为 2. 则与此正四面体的四个顶点距离相等的所有平面截这个四面体的外接球所得截面面积的总和为一个分数形式$\frac{m\pi}{n}$，请给出m + n的值。
Answer: 65
Problem: 已知 \( \sin \alpha+2 \cos \beta=2 \). 则\( y=\sin \left(\alpha+\frac{\pi}{4}\right)+2 \sin \left(\beta+\frac{\pi}{4}\right) \)的值域为 $\left[\sqrt{a}-\frac{\sqrt{b}}{c}, \sqrt{d}+\frac{\sqrt{e}}{f}\right]$.请提供a+b+c+d+e+f的值
Answer: 28
Problem: 已知\(f(x)=\log _{a} x(a>0, a \neq 1)\), \(g(x)=x^{2}-6 x+9\). 若存在唯一实数 \(x\), 使得 \(f(g(x))=g(f(x))=0\), 求实数 \(a\) 的值。标准答案格式为\sqrt[m]{n}，计算a所有取值的m+n的值的和
Answer: 12
Problem: 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。
Answer: 5
Problem: 已知双曲线$\frac{x^2}9-\frac{y^2}{16}=1$ 的左、右焦点分别为$F_1$、$F_2.$ 过$F_1$ 的直线与左支交于 $A$、$B$ 两点，且 $\frac{|AF_1|}{|AF_2|}=\frac12$ .则 $\triangle AF_1F_2$ 与$\triangle ABF_2$ 的内切圆面积之比为 $\qquad.$ 请给出比值两数之和。
Answer: 13
Problem: 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，前述结论不再成立。求道路条数的最大值，并请给出当n=2时的最大值。
Answer: 1
Problem: 在 $\triangle A B C$ 中, $O$ 为 $\triangle A B C$ 的外心，且满足 $\overrightarrow{A O} \cdot \overrightarrow{A B}+\overrightarrow{B O} \cdot \overrightarrow{B C}=\overrightarrow{C O} \cdot \overrightarrow{C A}$. 计算$\angle B$ 的大小，答案形式为 \frac{m\pi}{n}，请给出 m + n 的值。
Answer: 3
Problem: 已知数列 $\left\{a_{n}\right\},\left\{b_{n}\right\}$ 满足:
$a_{1}=a_{2}=2019, b_{1}=b_{2}=2017$ ，
且对于一切正整数 $n \geqslant 2$ ，均有
$$
\begin{aligned}
& a_{n+1}=\left(a_{n-1}-1\right)\left(b_{n}+1\right) \\
& b_{n+1}=a_{n} b_{n-1}-1
\end{aligned}
$$
求 $\sum_{n=1}^{\infty} b_{n}\left(\frac{1}{a_{n+1}}-\frac{1}{a_{n+3}}\right)$ 的值。原始答案是 \frac{m}{n} 的形式，请给出 m + n 的值。
Answer: 8148685
Problem: 已知正四棱锥的五个顶点在同一个球面上. 若该正四棱锥的体积为 9 ，则球的表面积的最小值为。原始答案为\frac{m\pi}{n}的形式，请给出m + n的值。
Answer: 85
Problem: 已知 $\theta$ 为 $g(x)=x^{4}-7 x^{2}+1$ 的一个零点, $h(x)$ 为一个有理系数的二次函数, 且满足 $h\left(\theta^{2}-1\right)=0$. 请给出 $h(x)=0$ 的根的形式为 $\frac{m \pm n\sqrt{k}}{p}$，请计算 m + n + k +p的值。
Answer: 15
Problem: 已知抛物线 $x^{2}=4 y$ 上的两个动点 $A\left(x_{1}, y_{1}\right) 、 B\left(x_{2}, y_{2}\right)\left(y_{1}+y_{2}=2, y_{1} \neq y_{2}\right)$ 。若线段 $A B$ 的垂直平分线与 $y$ 轴交于点 $C$, 则 $\triangle A B C$ 面积的最大值为某个复杂分数形式，标准答案格式为$\frac{a\sqrt{b} }{c}$，则a+b+c的值为多少。
Answer: 31
Problem: 已知 $a 、 b \in[1,3], a+b=4$ 。求 $f(a, b)=\left|\sqrt{a+\frac{1}{b}}-\sqrt{b+\frac{1}{a}}\right|$ 的最大值。原始答案为m-\frac{n}{\sqrt{k}}的形式，请给出m + n + k的值。
Answer: 7
Problem: Find all positive integer pairs $(x, y)$ such that $x^{2}-y^{119}=1$. Please provide the sum of all possible $x$ and $y$ values from the solutions.If the equation has no solution, then the answer is 0.
Answer: 0
Problem: 2020 个不同的小球放入 2020 个不同的盒子里，每个球落入各个盒子的可能性相同，不同小球之间落入盒子的可能性互不影响。则空盒子个数的期望为 \qquad 。原始答案为 \frac{m^{n}}{k^{q}} 格式，请给出 m + n +k+q的值。
Answer: 8078
Problem: $\frac{2020}{6063}$
Answer: 8083
Problem: 已知$x_{1},x_{2},\cdots,x_{50}\in\mathbf{R}$，且$\sum\limits_{i=1}^{50}\frac{x_{i}}{k+i}=\frac{2021}{2k+1}$对于任意$k\in\{1,2,\cdots,50\}$成立。求$\sum\limits_{i=1}^{50}\frac{x_{i}}{4042i+2021}$的值。原始答案为\(\frac{m}{n}\)格式，请给出m + n的值。
Answer: 12751
Problem: 已知关于$x$的不等式$$(2ax-\ln x)\left(x^{2}-(a+1)x+1\right)\geqslant0$$ 对于任意$x\in(0,+\infty)$恒成立。则实数$a$的取值范围是_______。标准答案格式为$\left[\frac{a}{be},c\right]$，计算a+b+c的值。
Answer: 4
Problem: 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 $A B C D$ ， $\angle A B C=\theta\left(\theta \in\left(0, \frac{\pi}{2}\right)\right). F$ 为 $P C$ 的中点，过 $F$ 且与平面 $P A B$ 平行的平面 $\alpha$ 分别与 $B C$ 、 $P D$ 交于点 $E 、 H$. 若 $A B=A C, P A=A D=4$, 则点 $F$ 到平面 $A E H$ 的距离为\frac{\sqrt{m}}{n}的形式，请给出m + n的值。
Answer: 4
Problem: 设$a_{1},a_{2},	ext{...},a_{2021}$均为非负实数，满足$a_{1}+a_{2}+	ext{...}+a_{2021}=1$。定义：$S=	ext{...}rac{a_{i}}{i},T=	ext{...}rac{a_{i}a_{j}}{ij}$。求$S+2T$的最小可能值和最大可能值。原始答案为两个分数，请给出这些分数中所有分子和分母的和。
Answer: 2055
Problem: 在 $\triangle ABC$ 中，$BC=2\sqrt{2}$，$AC=\sqrt{6}-\sqrt{2}$，$\angle BAC=\angle ABC+30^{\circ}$。求 $S_{\triangle ABC}$ 的整数值，假设原始答案为 $a-\sqrtb$ 的形式，请给出 $a+b$ 的整数值。
Answer: 6
Problem: 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 $A-P B Q$ 的体积最大时， $C E$ 的原始答案为 \(\dfrac{\sqrt{m}}{n}\) 的形式，请给出 m + n 的值。
Answer: 8
Problem: 已知复数 $z$ 满足 $|z|=3$. 则 $\left|\frac{z^{2}+1}{z^{2}+\mathrm{i} z+z+\mathrm{i}}\right|$ 的最大值为原始答案的形式为\(\dfrac{m\sqrt{n}+\sqrt{k}}{q}\)，请给出m + n + k+q的值。
Answer: 95
Problem: 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}{n_1}和\dfrac{m_2}{n_2}格式，请给出m_1+n_1+m_2+n_2的值。
Answer: 8
Problem: 设 $a_{1}, a_{2}, \cdots, a_{2021}$ 为 2021 个整数，满足 $1=a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{2021}=100$ 。记  
$$  
\begin{aligned}  
f= & \left(a_{1}^{2}+a_{2}^{2}+\cdots+a_{2021}^{2}\right)- \\
& \left(a_{1} a_{3}+a_{2} a_{4}+\cdots+a_{2019} a_{2021}\right).  
\end{aligned}  
$$  
求 $f$ 的最大值 $f_{0}$, 并求使得 $f=f_{0}$ 成立的数组 $\left(a_{1}, a_{2}, \cdots, a_{2021}\right)$ 的个数。请给出 $f_0$ 和数组个数之和。
Answer: 19901
Problem: 当$m=3$ 时，则使得 $m^{2}+n$ 整除 $n^{2}+m$ 的最大正整数 $n$ 是 $\qquad$。
Answer: 5
Problem: 设 $[x]$ 表示不超过实数 $x$ 的最大整数。则函数 $f(x)=\frac{x+1}{[x]+1}$ 的值域是一个区间。请给出该区间两个端点的和。
Answer: 2
Problem: 在平面直角坐标系 $x O_{y}$ 中, 椭圆 $\Gamma_{1}:$ $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ 和抛物线 $\Gamma_{2}: y^{2}=\frac{1}{2} a x$交于 $A 、 B$ 两点， $P$ 为 $\Gamma_{1}$ 的右顶点。若 $O 、 A 、 P 、$ $B$ 四点共圆, 则 $\Gamma_{1}$ 的离心率的$\sqrt{6}$倍是 $\qquad$。
Answer: 2
Problem: 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 $A B_{1} \dot{D}$ 。以 $D$ 为球心、 $\frac{\sqrt{7}}{2}$ 为半径的球面与侧面 $A A_{1} B_{1} B$ 的交线长的$\frac{3}{\pi}$倍是 $\qquad$.
Answer: 2
Problem: 给定正整数 $n$, 非负实数 $a_{1}$, $a_{2}, \cdots, a_{n}$ 满足 $a_{1}+a_{2}+\cdots+a_{n}=4$ 。记
$$
S=54(a_{1}+a_{1} a_{2}+\cdots+a_{1} a_{2} \cdots a_{n})-13\sqrt{13}.
$$
求 $S$ 的最大值.
Answer: 305
Problem: 求所有的正整数 $n$ ，存在一种方案将集合 $\\{1,2, \\cdots, 2 n\\}$ 划分为元素个数为 2 且两两不交的集合 $A_{1}, A_{2}, \\cdots, A_{n}$ ，使得不存在 $x_{1}, x_{2}, \\cdots, x_{n}$ 同时满足:（1） $x_{i} \\in A_{i}(i=1,2, \\cdots, n)$;(2) $2 n \\mid\\left(x_{1}+x_{2}+\\cdots+x_{n}\\right)$. 请给出满足条件的n的最小值。
Answer: 2
Problem: 求正整数 $a, b$ 应满足的条件，使得能够将集合 $\{1,2, \cdots, a b\}$ 分成 $a$ 个不相交的 $b$ 元子集，满足每个子集的元素和都相等。请给出所有满足条件的(a, b)对中a + b的最小值。
Answer: 4
Problem: 已知数列 $\left\{a_{n}\right\}$ 满足:
$$
a_{1}=1, a_{2}=3,
$$
$a_{n}= \begin{cases}a_{\frac{n}{}}^{2}+2, & n \text { 是偶数; } \\ 2 a_{\frac{n+1}{2}}-1, & n \text { 是奇数. }\end{cases}$
当$n=6$，求数列 $\left\{a_{n}\right\}$ 的前 $2^{n}$ 项和 $S_{2^{n}}$.
Answer: 1394
Problem: 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 $M N$ 垂直, 并交直线 $x=4$ 于点 $P$, 直线 $O P$ 交 $\Gamma$ 于 $A 、 B$ 两点. 设直线 $M P 、 M N 、 M A 、 M B$ 的斜率依次为 $m 、 n 、 p$ 、 $q$ (四条直线的斜率均存在且均不为 0 ). 记 $m+n=s, p+q=t, m n=r, s t=f(r)$. 求 $f(\frac{3}{4})$。
Answer: 0
Problem: 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 $\qquad$。
Answer: 2
Problem: 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, $C D=3, \angle A O C=\theta \in\left[\frac{\pi}{3}, \frac{5}{6} \pi\right]$, 四面体 $A B C D$的体积的最大值记为 $V(\theta)$. 则 $4\sqrt{3}V(\theta)$ 的最小值是 $\qquad$。
Answer: 27
Problem: 已知向量 $\boldsymbol{a} 、 \boldsymbol{b} 、 \boldsymbol{c}$ 满足 $|\boldsymbol{a}|=|\boldsymbol{b}|=|\boldsymbol{c}|=4$, $\boldsymbol{a} \cdot \boldsymbol{b}=8$. 则 $|3 \boldsymbol{c}-\boldsymbol{a}|+\left|\frac{1}{4} \boldsymbol{c}-\boldsymbol{b}\right|$ 的最小值的$\sqrt{133}$倍是 $\qquad$.
Answer: 133
Problem: 给定正整数 $6$ 。求最大的实数 $M$ ，使得对于任意的实数 $x_{i}(1 \leqslant i \leqslant 2 n+1)$ ，若 $x_{1} \leqslant x_{2} \leqslant \cdots \leqslant x_{2 n+1}$, 则 $\sum_{i=1}^{2 n+1} x_{i}^{2} \geqslant M x_{n+1}^{2}$.
Answer: 7
Problem: 函数 $f(x)=8\sqrt{15}(7 \sin x+\sin 2 x)$ 的最大值为 $\qquad$
Answer: 225
Problem: 已知函数  \(f(x)=3 \lg (x+2)-\log _{2} x\),集合 \(M=\left\{n \in \mathbf{Z} \mid f\left(n^{3}-3 n^{2}-10 n+32\right) \geqslant \mathbf{0}\right\}\) 。求 \(M\) 中所有元素的和。
Answer: 6
Problem: 已知 $n$ 为给定整数且 $n \geqslant 2$ ，集合 $A$ 的元素均为正整数， $A$ 的最小元素为 1 ，最大元素为 $a$ ，且 $7 \times 3^{n}<a<3^{n+2}$ ，满足对于 $A$ 的任一元素 $x(x \neq 1)$ ，均存在 $s 、 t 、 p \in$ $A$ ，使得 $x=s+t+p$ 。求集合 $A$ 元素个数的最小值，并给出 n=2 时的答案。
Answer: 6
Problem: 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
Answer: 1
Problem: 已知$f(x)$为定义在$\mathbf{R}$上的奇函数，当$x\leqslant0$时，
$$f(x)=\left|x+\frac{1}{2}\sin\alpha\right|+\left|x+\frac{3}{2}\sin\alpha\right|-2\sin\alpha$$
其中，$\alpha\in(-\pi,\pi)$。若对任意实数$x$，均有$f(x)\geqslant f(x+2)$，则$\alpha$的最小值
Answer: 0
Problem: 三棱雉$P-ABC$中，三个侧面与底面所成角相等，三个侧面的面积分别为$3、4、5$，且底面面积为6。已知三棱雉$P-ABC$的外接球的表面积的原始答案形式为 $\frac{m\pi}{n}$，请给出 m + n 的值。
Answer: 82
Problem: 已知实数$x、y$满足：
$x>\max\{-3,y\},(x+3)\left(x^{2}-y^{2}\right)=8$。
则$3x+y-4\sqrt6$的最小值为_______。
Answer: -6
Problem: 一只小虫在正八面体的表面上爬行，每秒从某一个顶点等可能地爬往四个相邻顶点之一。则小虫在第8秒爬回初始位置的概率的256倍为_______。
Answer: 43
Problem: 已知复数 $z_{1} 、 z_{2} 、 z_{3}$ 的辐角分别为 $\alpha 、 \beta 、 \gamma$, 且满足:  $$  \left|z_{1}\right|=1,\left|z_{2}\right|+\left|z_{3}\right|=2, z_{1}+z_{2}+z_{3}=0  $$  求 $\cos (\alpha-\beta)+2 \cos (\beta-\gamma)+3 \cos (\gamma-\alpha)$的取值范围。请给出该范围的起点和终点的和。
Answer: -4
Problem: 设 $\alpha$ 为复数， $\bar{\alpha}$ 表示 $\alpha$ 的共轭复数。已知 $|\alpha-\bar{\alpha}|=2 \sqrt{3}$ 且 $\frac{\alpha}{\bar{\alpha}^{2}}$ 为纯虚数，则 $\sqrt{3}|\alpha|$ 的所有可能值的和为 $\qquad$.
Answer: 3
Problem: 已知正数 $a, b, c$, 满足 $(a+b-c)\left(\frac{1}{a}+\right.$ $\left.\frac{1}{b}-\frac{1}{c}\right)=3$, 求 $f=\left(a^{4}+b^{4}+c^{4}\right)\left(\frac{1}{a^{4}}+\frac{1}{b^{4}}+\frac{1}{c^{4}}\right)-240\sqrt{3}$ 的最小值.
Answer: 417
Problem: 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 $D$ 到 $E(4,5)$ 距离最大值和最小值之积减去$2\sqrt{170}-2\sqrt{85}$为 $\qquad$.
Answer: 0
Problem: 函数 $f(x)=\min \left\{\sin x, \cos x,-\frac{1}{\pi} x+1\right\}$ 在 $[0, \pi]$ 上最大值的$\sqrt{2}$倍是 $\qquad$.
Answer: 1
Problem: 解方程组 $\left\{\begin{array}{l}x y=2 x+y-1 \\ x z=3 x+4 z-8 \\ y z=3 y+2 z-8\end{array}\right.$中x所有的可能值之和。
Answer: 5
Problem: 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。
Answer: 5
Problem: 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。
Answer: 8
Problem: 求 $f(x)=711(|x-1|+|2 x-1|+\cdots+|2011 x-1|)$ 的最小值.
Answer: 592043
Problem: 设整数 $n \geq 2$. 某天 $n$ 个人组成了一个社团，每人向政府交 1 元注册费。接下来每天都发生如下事情：每一个前一天组建的社团都分裂成两个团体，其中含至少两人的团体组建成一个新社团，每个新社团的成员向政府交 1 元注册费，只有一名成员的团体不再组建新社团。上述过程直到不能继续进行下去为止。当$n=6$时，求政府总共收到的注册费的最大值。
Answer: 20
Problem: 数列 $\left\{a_{n}\right\}$ 满足  
$$  
a_{0}=1, a_{n+1}=\sum_{\substack{i, j, k \geq 0 \\ i+j+k=n}} a_{i} a_{j} a_{k}  
$$  
对任意 $n \geq 0$ 成立. 求数列 $\left\{a_{n}\right\}$ 的通项. 原始答案为\frac{m}{n}格式，请给出m + n的值，当n = 2时。
Answer: 3
Problem: 已知实数 $x, y$ 满足 $\left\{\begin{array}{l}\sin x+\sin y=\frac{1}{3} \\ \cos x-\cos y=\frac{1}{5}\end{array}\right.$, 求 $17\sin (x-y)$.
Answer: -15
Problem: 函数 $f(x)=\frac{\sqrt{2}}{2}(\cos x-\sin x) \sin \left(x+\frac{\pi}{4}\right)-2 a \sin x+b(a>0)$ 的最大值为 1, 最小值为 -4, 求 $b$ 的值.
Answer: -1
Problem: 已知椭圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ 与圆 $x^{2}+y^{2}=b^{2}$, 过椭圆上一点 $M$ 作圆的两切线, 切点分别为 $P, Q$, 直线 $P Q$ 与 $x, y$轴分别交于点 $E, F$, 当$a=2,b=2$时，求 $S_{\triangle E O F}$ 的最小值.
Answer: 4
Problem: 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 $\lim _{x \rightarrow+\infty} 2\sqrt{3}y=$
Answer: 3
Problem: Evaluate $29 \frac{27}{28} \times 27\frac{14}{15}$. The original answer is in the form of $a + \frac{m}{n}$, where a is the integer part. Please provide the value of a + m + n.
Answer: 1358
Problem: During a rebuilding project by contractors 'A', 'B' and 'C', there was a shortage of tractors. The contractors lent each other tractors as needed. At first, 'A' lent 'B' and 'C' as many tractors as they each already had. A few months later, 'B' lent 'A' and 'C' as many as they each already had. Still later, 'C' lent 'A' and 'B' as many as they each already had. By then each contractor had $24$ tractors. Please provide the total number of tractors the contractors originally had.
Answer: 204
Problem: Find the two $10$-digit numbers which become nine times as large if the order of the digits is reversed. Please provide the sum of these two numbers.
Answer: 1287128711
Problem: In a language college, $72\%$ students can speak Chinese, $65\%$ students can speak English, and $10\%$ students can speak neither Chinese nor English. Find the percentage of students who can speak both Chinese and English. Please provide the sum of the original percentage and 100.
Answer: 147
Problem: Replace the letters $a$, $b$, $c$ and $d$ in the following expression with the numbers $1$, $2$, $3$ and $4$, without repetition: $a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d}}}$ Find the difference between the maximum value and the minimum value of the expression. The original answer is in \frac{m}{n} format, please give the value of m + n.
Answer: 67
Problem: Skipper’s doghouse has a regular hexagonal base that measures one metre on each side. Skipper is tethered to a $2$-metre rope which is fixed to a vertex. What is the area of the region outside the doghouse that Skipper can reach? Calculate an approximate answer by using $\pi=3.14$ or $\pi=22/7$. The original answer is in the format $n\pi$, please give the value of n.
Answer: 3
Problem: Find all pairs of primes $(p,q)$ for which $p-q$ and $pq-q$ are both perfect squares. Please provide the value of $p + q + pq$.
Answer: 11
Problem: Find all triples of positive integers $(x,y,z)$ that satisfy the equation  

$$\begin{align*} 2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023 \end{align*}$$

Please provide the sum of all elements in one such triple.
Answer: 6