ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: $\sum\limits_{k=0}^{2022} C_{2022}^{k} \cos \frac{(1011-k) \pi}{2}$ 的值为$a^{b}$，求$a+b$的值。
Answer: 1013
Problem: 已知函数$f(x)=2\left(\sin ^{4} x+\cos ^{4} x\right)+m(\sin x+\cos x)^{4}$在 $x \in\left[0, \frac{\pi}{2}\right]$ 有最大值 5。请计算$m$的值加上5后的结果。
Answer: 6
Problem: For a positive integer $n$, let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$. Find the smallest $n$ such that $\left|f_n^{\prime \prime}(0)\right|>2023$. Please provide the value of $n$.
Answer: 18
Problem: An unbiased coin is tossed $n$ times. $H$ is the number of heads and $T$ is the number of tails. What is the expected value of $|H-T|$? In other words, evaluate in closed form:
$$
\frac{1}{2^{n-1}} \sum_{k<n / 2}(n-2 k)\binom{n}{k}
$$
(In this problem, "closed form" means a form that does not involve a series. The given series can be reduced to a single term involving only binomial coefficients, rational functions of $n$ and $2^{n}$, and the largest integer function $[x]$.)
The expected value of $|H-T|$ is $\frac{b}{a}$ when $n=3$. Find the value of $a+b$.
Answer: 5
Problem: For each $t\geqslant0$, let $S_{t}$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$ f(1)-2 f(2 / 3)+f(1 / 3)\geqslant t[f(2 / 3)-2 f(1 / 3)+f(0)] . $$ Develop necessary and sufficient conditions on $t$ for $S_{t}$ to be closed under multiplication. Please provide the sum of the endpoints of the interval for which the condition holds.
Answer: 1
Problem: for all polygons $P$, where the double integral is taken over $H(P)$. The original answer is in $\frac{m}{n}$ format, please give the value of $m + n$.
Answer: 11
Problem: A convex pentagon $P = ABCDE$, with vertices labeled consecutively, is inscribed in a circle of radius 1. Find the maximum area of P subject to the condition that the chords AC and BD be perpendicular. The original answer is in the format $1 + \frac{m}{n} \sqrt{3}$, please give the value of $m + n$.
Answer: 7
Problem: Assume $F$ is a finite field with an odd number of elements $m$. Assume $p(x)$ is an irreducible (i.e., non-factorizable) polynomial in $F$ of the form
$$
x^{2}+b x+c, \quad b, c \in F
$$
For how many elements $k$ in $F$, $p(x)+k$ is irreducible in $F$? How many elements are there when $m=7$.
Answer: 3
Problem: Let $u, f$, and g be functions defined for all real numbers x, such that

$$
\frac{u(x+1)+u(x-1)}{2}=f(x) \text { and } \frac{u(x+4)+u(x-4)}{2}=g(x).
$$

If $u(x)=g(x+a)-f(x+b)+f(x+c)-f(x+d)+f(x+e)$, what is the value of $a+b+c+d+e$.
Answer: 20
Problem: Evaluate the infinite product
$$
\prod_{n=2}^{\infty} \frac{n^{3}-1}{n^{3}+1}
$$
The original answer is in \frac{m}{n} format, please give the value of m + n.
Answer: 5
Problem: Determine all polynomials $P(x)$ such that $P\left(x^{2}+1\right)=(P(x))^{2}+1$ and $P(0)=0$. Please provide the minimum positive integer value $P (x) $that meets the criteria.
Answer: 1
Problem: Establish necessary and sufficient conditions on the constant $k$ for the existence of a continuous real valued function $f(x)$ satisfying $f(f(x))=k x^{9}$ for all real x. Provide the smallest integer value of k that satisfies the condition.
Answer: 0
Problem: For each $t\geqslant0$, let $S_t$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$f(1)-2f(2/3)+f(1/3)\geqslant[f(2/3)-2f(1/3)+f(0)]$$. Develop necessary and sufficient conditions on $t$ for $S_t$ to be closed under multiplication. Please provide the value of a + b if the original answer is in the format $a\leqslant t\leqslant b$.
Answer: 1
Problem: Let r and s be positive integers. Derive a formula for the number of ordered quadruples $(a, b, c, d)$ of positive integers such that
$$3^{r}·7^{s}=
\mathrm{lcm}[a,b,c]=\mathrm{lcm}[a,b,d]=\mathrm{lcm}[a,c,d]=\mathrm{lcm}[b,c,d]$$.
Please provide the value of the formula when r = 2 and s = 1. The answer should be the sum of the coefficients of the expanded expression of the original answer.
Answer: 1089
Problem: Find positive integers $n$ and $a_{1}, a_{2}, \ldots, a_{n}$ such that \n$$\na_{1}+a_{2}+\cdots+a_{n}=1979\n$$\nand the product $a_{1} a_{2} \cdots a_{n}$ is as large as possible. Please provide the value of $a + n$.
Answer: 662
Problem: Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $0<x<1$ with $f(0)=0$ and $f(1)=1$. Determine the largest real number $u$ such that
$$
u \leqslant \int_{0}^{1}\left|f^{\prime}(x)-f(x)\right| d x$$
for all $f$ in $C$. The original answer is in \(\frac{m}{ne}\) format, please provide the value of m+ n.
Answer: 2
Problem: A point P moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection. Let $N(T)$ be the number of starting directions from a fixed interior point $P_{0}$ for which P escapes within T units of time. Find the least constant a for which constants b and c exist such that $$ N(T) \leq a T^{2}+b T+c $$ for all $T>0$ and all initial points $P_{0}$. The original answer is in \(\pi\) format, please give the integer part of the answer.
Answer: 1
Problem: Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. The answer is in the form $r+s \sqrt{t}$ with $r, s$, and $t$ being positive integers. Please provide the value of $r + s + t$.
Answer: 27
Problem: Let $p$ be a prime number. Let $J$ be the set of all $2 \times 2$ matrices $\binom{a b}{c d}$ whose entries are chosen from $\{0,1,2, \cdots, p-1\}$ and satisfy the conditions $a+d \equiv 1(\bmod p), a d-b c \equiv 0(\bmod p)$. Determine how many members $J$ has. The original answer is in the format of $p^2 + p$. Let p=5，Determine how many members $J$ has.
Answer: 30
Problem: Evaluate $$\sum_{k=0}^{n} (-1)^k\binom{n}{k} (x-k)^n.$$ Provide the integer value of n=3.
Answer: 6
Problem: Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i](i=1, 2,3)$. If the distribution of each random number is uniform with respect to length in the interval it is chosen from, determine the expected value of the smallest of the three numbers chosen. The original answer is in the format $\frac{a}{b} L_{1}^{2} L_{2} L_{3}-\frac{c}{d} L_{1}^{3}\left(L_{2}+L_{3}\right)+\frac{e}{f} L_{1}^{4}$, please give the value of a+b+c+d+e+f.
Answer: 23
Problem: Let $F(x)$ be a real valued function defined for all real $x$ except for $x=0$ and $x=1$ and satisfying the functional equation $F(x)+F\{(x-1) / x\}=1+x$. Find all functions $F(x)$ satisfying these conditions. The original answer is in the format \frac{m}{n}, please provide the value of m + n when x = 2.
Answer: 7
Problem: A particle moving on a straight line starts from rest and attains a velocity $v_0$ after traversing a distance $s_0$. If the motion is such that the acceleration was never increasing, find the maximum time for the traverse. 假设So=2,Vo=1,The original answer is in the format \frac{m}{n}, please give the value of m+n.
Answer: 5
Problem: Let u, f, and g be functions, defined for all real numbers x, such that $$\frac{u(x+1)+u(x-1)}{2}=f(x) \text { and } \frac{u(x+4)+u(x-4)}{2}=g(x).$$ Determine u(x) in terms of f and g. Please provide the value of u(x) when x = 1. The answer should be in the format $u(x)=g(a)-f(b)+f(c)-f(d)+f(e)$, directly give the value of a+b+c+d+e.
Answer: 25
Problem: Determine the minimum value of
$$(r-1)^{2}+\left(\frac{s}{r}-1\right)^{2}+\left(\frac{t}{s}-1\right)^{2}+\left(\frac{4}{t}-1\right)^{2}$$
for all real numbers r, s, t with $1 \leq r \leq s \leq t \leq 4$.The original answer form is $a-b \ sqrt {c} $, find the value of a+b+c
Answer: 22
Problem: Let $ABC$ be a triangle with angle $A< \angle C<90^\circ < \angle B$. Consider the bisectors of the external angles at $A$ and $B$, each measured from the vertex to the opposite side (extended). Suppose both of these line-segments are equal to $AB$. Compute the sum of angles $A$ and $B$. 请给出角A和角B的和。
Answer: 24
Problem: (a) Find a solution that is not identically zero, of the homogeneous linear differential equation $$( 3x^2+x-1 ){y}''- ( 9x^2+9x-2 ){y}'+( 18x+3 )y=0$$ Intelligent guessing of the form of a solution may be helpful. 
(b) Let $y=f(x)$ be the solution of the nonhomogeneous differential equation $$( 3x^2+x-1){y}''-( 9x^2+9x-2){y}'+( 18x+3)=6( 6x+1) $$ that has $f(0)=1$ and $(f(-1)-2)(f(1)-6)=1$. Find integers $a, b, c$ such that $(f(-2)-a)(f(2)-b)=c$. Please provide the value of a + b + c.
Answer: 21
Problem: Assume that the differential equation $$y^{\prime\prime\prime}+p(x)y^{\prime\prime}+q(x)y^{\prime}+r(x)y = 0$$ has solutions $y_1(x)$, $y_2(x)$, and $y_3(x)$ on the whole real line such that $$y_{1}^{2}(x)+y_{2}^{2}(x)+y_{3}^{2}(x)=1$$ for all real x. Let $$f(x)=\left(y_{1}^{\prime}(x)\right)^{2}+\left(y_{2}^{\prime}(x)\right)^{2}+\left(y_{3}^{\prime}(x)\right)^{2}$$. Find constants $A$ and $B$ such that $f(x)$ is a solution to the differential equation $$y^{\prime}+Ap(x)y=Br(x)$$. The original answer is in the form of \frac{m}{n}. Please provide the value of m + n.
Answer: 5
Problem: A sequence of convex polygons $\{P_n\}$, $n\geqslant0$, is defined inductively as follows. $P_0$ is an equilateral triangle with sides of length 1. Once $P_n$ has been determined, its sides are trisected; the vertices of $P_{n+1}$ are the interior trisection points of the sides of $P_n$. Thus, $P_{n+1}$ is obtained by cutting corners off $P_n$, and $P_n$ has $3\cdot2^n$ sides. ($P_1$ is a regular hexagon with sides of length $1/3$.) Express $\lim_{n\rightarrow\infty}Area(P_n)$ in the form $\sqrt{a}/b$, where a and b are positive integers. Please provide the value of a + b.
Answer: 10
Problem: Let $V$ be the region in the cartesian plane consisting of all points $(x, y)$ satisfying the simultaneous conditions $$|x|\leqslant y\leqslant|x|+3\text{ and }y\leqslant4.$$ Find the centroid $(\bar{x},\bar{y})$ of $V$. The original answer for $\bar{y}$ is in the form \(\frac{m}{n}\). Please provide the value of  m + n$.
Answer: 18
Problem: (a) Let $z$ be a solution of the quadratic equation
$$
a z^{2}+b z+c=0
$$
and let $n$ be a positive integer. Show that $z$ can be expressed as a rational function of $z^{n}, a, b, c$.
(b) Using (a) or by any other means, express $x$ as a rational function of $x^{3}$ and $x+(1 / x)$. The original answer is in \frac{m}{n} format, please give the value of m + n if x=v=2.
Answer: 13
Problem: Express
$$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n}$$
as a rational number. The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
Answer: 11
Problem: Evaluate $$\lim\limits_{x\to1^-}\prod\limits_{n=0}^\infty\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}.$$ Please provide the value of a-b, where the original answer is in the form of \ln a-b.
Answer: 1
Problem: Determine all real numbers $a>0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region $$R=\{(x,y);0\leq x\leq a,0\leq y\leq f(x)\}$$ has perimeter $k$ units and area $k$ square units for some real number $k.$ Please provide the smallest integer value of $a$ that satisfies the condition.
Answer: 2
Problem: Evaluate $\int_{0}^{1}\frac{ln(x+1)}{x^2+1}dx$. The original answer is in the format $\frac{m\pi}{n}$, please provide the value of $m + n$.
Answer: 10
Problem: Let $S={(a, b) \mid a=1,2,..., n, b=1,2,3}$. A rook tour of $S$ is a polygonal path made up of line segments connecting points $p_1, p_2,\dots,p_{3n}$ in sequence such that  (i) $p_i \in S$,  (ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1 \le i< 3n$,  (iii) for each $p\in S$ there is a unique i such that $p_i =p$. How many rook tours are there that begin at $(1,1)$ and end at $(n,1)$? Please provide the value of the number of rook tours when n = 5.
Answer: 8
Problem: Let $a_1, a_2, \ldots, a_n$ be real numbers, and let $b_1, b_2, \ldots, b_n$ be distinct positive integers. Suppose there is a polynomial $f(x)$ satisfying the identity $$(1-x)^n f(x)=1+\sum_{i=1}^n a_i x^{b_i}.$$ Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \ldots, b_n$ and $n$ (but independent of $a_1, a_2, \ldots, a_n$). Provide the value of $b_1 + b_n + n$ for $b_1 = 2$, $b_n = 3$, and $n = 2$.
Answer: 7
Problem: Evaluate $\sum_{n=0}^{\infty} \operatorname{Arccot}\left(n^2+n+1\right)$, where $\operatorname{Arccot} t$ for $t \geqslant 0$ denotes the number $\theta$ in the interval $0<\theta \leqslant \pi / 2$ with $\cot \theta=t$. The original answer is in the format \frac{m\pi}{n}, please give the value of m + n.
Answer: 3
Problem: Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral. The original answer is in the format $1-\frac{m}{n\pi^2}$, please provide the value of m + n.
Answer: 47
Problem: Find the volume of the region of points $(x,y,z)$ such that $$(x^2+y^2+z^2+8)^2\leq36(x^2+y^2).$$ The original answer is in the format $m\pi^{n}$, please give the value of m + n.
Answer: 8
Problem: For each continuous function $f:[0,1]\to\mathbb{R}$, let $I(f)=$ $\int _{0}^{1}x^{2}f( x)$ $dx$ and $J( x) = \int _0^{1}x\left ( f( x) \right ) ^{2}$ $dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$ The original answer is in the format \frac{m}{n}, please give the value of m + n.
Answer: 17
Problem: Let $k$ be an integer greater than 1. Suppose $a_0>0$, and define $$a_{n+1}=a_n+\frac{1}{\sqrt[k]{a_n}}$$ for $n>0.$ Evaluate $$\lim_{n\to\infty}\frac{a_n^{k+1}}{n^k}.$$ The original answer is in \left(\frac{k+1}{k}\right)^{k} format, please give the value of m + n where m/n is the simplified fraction form of the original answer when k = 1.
Answer: 2
Problem: Let $n$ be a positive integer. Find the number of pairs $P,Q$ of polynomials with real coefficients such that $$(P(X))^2+(Q(X))^2=X^{2n}+1$$ and $\deg P>\deg Q.$ Provide the value when $n=3$. The answer should be an integer.
Answer: 16
Problem: What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1? The original answer is in the format $\frac{\sqrt{m}}{n}$, please provide the value of m + n.
Answer: 4
Problem: 求出所有的正整数 $n$, 使得关于 $x 、 y$的方程 $$ \frac{1}{x}+\frac{1}{y}=\frac{1}{n} $$ 恰有 2011 组满足 $x \leqslant y$ 的正整数解 $(x, y)$。已知答案为p^{m}的形式，给出m 的值。
Answer: 2010
Problem: 给定正整数 $n \geqslant 2$ ，设方格表  
$$  
\begin{aligned}  
& A=\left(\begin{array}{cccc}  
a_{11} & a_{12} & \cdots & a_{1 n} \\  
a_{21} & a_{22} & \cdots & a_{2 n} \\  
\vdots & \vdots & \vdots & \vdots \\  
a_{n 1} & a_{n 2} & \cdots & a_{n n}  
\end{array}\right), \\  
& B=\left(\begin{array}{cccc}  
b_{11} & b_{12} & \cdots & b_{1 n} \\  
b_{21} & b_{22} & \cdots & b_{2 n} \\  
\vdots & \vdots & \vdots & \vdots \\  
b_{n 1} & b_{n 2} & \cdots & b_{n n}  
\end{array}\right)  
\end{aligned}  
$$  
满足 $\left\{a_{i j} \mid 1 \leqslant i 、 j \leqslant n\right\}=\left\{b_{i j} \mid 1 \leqslant i 、 j \leqslant n\right\}$  
$$  
=\left\{1,2, \cdots, n^{2}\right\}  
$$  
对 $A$ 可以进行如下操作：选取位于同一 行或同一列的两个数，交换它们的位置，其余 $n^{2}-2$ 个数保持不动。这种操作称为一次对换. 求最小正整数 $m$ ，使得对任意的 $A 、 B$ ，可以经过不超过 $m$ 次对换，把 $A$ 变成 $B$ 。请给出当 n = 2 时的 m 值。
Answer: 4
Problem: 求所有的三元正整数组 $(a, b, c)$ 满足 $\frac{a}{2^{a}}=\frac{b}{2^{b}}+\frac{c}{2^{c}}$ 。请给出所有满足条件的三元组的元素乘积之和。
Answer: 288
Problem: 设实数 $x_{1}, x_{2}, \cdots, x_{22}$ 满足对任意 $1 \leqslant i \leqslant$ 22, 有 $2^{i-1} \leqslant x_{i} \leqslant 2^{i}$ 。 求
$$
\left(x_{1}+x_{2}+\cdots+x_{22}\right)\left(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{22}}\right)
$$
的最大值。原始答案为$(2^{12}-1-\frac{1}{2^{11}})^{2}$，请给出近似整数值。
Answer: 16769021
Problem: 设平面上有 $n(n \geqslant 4)$ 个点 $V_{1}, V_{2}, \cdots$, $V_{n}$ ，任意三点不共线，某些点之间连有线段。把标号分别为 $1,2, \cdots, n$ 的 $n$ 枚棋子放置在这 $n$ 个点处，每个点处恰有一枚棋子。现对这 $n$ 枚棋子进行如下操作：每次选取若干枚棋子，将它们分别移动到与自己所在点有线段相连的另一个点处；操作后每点处仍恰有一枚棋子，并且没有两枚棋子在操作前后交换位置。若一种连线段的方式使得无论开始时如何放置这 $n$ 枚棋子，总能经过有限次操作
后，使每个标号为 $k(k=1,2, \cdots, n)$ 的棋子在点 $V_{k}$ 处，则称这种连线段的方式为 ""和谐的"". 求在所有和谐的连线段的方式中，当$n=3$时，线段数目的最小值。
Answer: 4
Problem: 给定整数 $n(n \geqslant 3)$ ，设 $A_{1}, A_{2}, \cdots, A_{2 n}$是集合 $\{1,2, \cdots, n\}$ 的两两不同的非空子集，记 $A_{2 n+1}=A_{1}$. 求  当$n=7$时，
$$  
\sum_{i=1}^{2 n} \frac{\left|A_{i} \cap A_{i+1}\right|}{\left|A_{i}\right| \cdot\left|A_{i+1}\right|}  
$$  
的最大值.
Answer: 7
Problem: 给定正整数 $n(n \geqslant 3)$ 。对于 $1,2, \cdots, n$的任意一个排列 $P=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$, 若 $i<j<k$, 则称 $x_{j}$ 介于 $x_{i}$ 和 $x_{k}$ 之间（如在排列（ $1,3,2,4$ ）中，3 介于 1 和 4 之间， 4 不介于 1 和 2 之间）。设集合 $S=\left\{P_{1}, P_{2}, \cdots, P_{m}\right\}$的每个元素 $P_{i}$ 是 $1,2, \cdots, n$ 的排列. 已知 \{1,2, \cdots, n\} 的任意三个不同数中都有一个数，它在每个 $P_{i}(1 \leqslant i \leqslant m)$ 中都不介于另外两个数之间。求 $m$ 的最大值。请给出当 $n = 5$ 时，$m$ 的最大值。
Answer: 16
Problem: 求所有的整数 $k$, 使得存在正整数 $a$和 $b$, 满足  $$  \frac{b+1}a+\frac{a+1}b=k.$$ 请给出所有满足条件的整数 $k$ 的和。
Answer: 7
Problem: 给定正整数 $n$ ，设 $a_{1}, a_{2}, \cdots, a_{n}$ 为非负整数序列，若其中连续若干项（可以只有一项）的算术平均值不小于 1 ，则称这些项组成一条"龙"，其中第一项称为"龙头"，最后一项称为"龙尾"。已知 $a_{1}, a_{2}, \cdots, a_{n}$ 中每一项均为龙头或者龙尾。求 \sum_{i=1}^{n} a_{i}$的最小值。请给出该值在n = 2时的结果。
Answer: 2
Problem: 定义 $n$ 元整数组的一次变换为  
$$  
\begin{aligned}  
& \left(a_{1}, a_{2}, \cdots, a_{n-1}, a_{n}\right) \\  
& \rightarrow\left(a_{1}+a_{2}, a_{2}+a_{3}, \cdots, a_{n-1}+a_{n}, a_{n}+a_{1}\right)  
\end{aligned}  
$$  
求所有的正整数对 $(n, k)(n 、 k \geqslant 2)$ ，满足对任意的 $n$ 元整数数组 $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ ，在有限次变换后所得数组中每一个数均为 $k$ 的倍数。请给出最小的n和k的和。
Answer: 4
Problem: 已知 $A=\left\{x \mid x^{2}-4 x+3<0, x \in R\right\}$,  
$$  
B=\left\{x \mid 2^{1-x}+a \leqslant 0, \quad x^{2}-2(a+7) x+5 \leqslant 0, \quad x \in \mathbf{R}\right\}  
$$  
若 $A \subseteq B$ ，则实数 $a$ 的取值范围是 。请给出该区间两个端点的和。
Answer: -5
Problem: 将关于 $x$ 的多项式 $f(x)=1-x+x^{2}-x^{3}+\cdots-x^{19}+x^{20}$ 表为关于 $y$ 的多项式 $g(y)=a_{0}+a_{1} y+a_{2} y^{2}+\cdots$ $+a_{19} y^{19}+a_{20}{ }^{20}$, 其中 $y=x-4$, 则 $6(a_{0}+a_{1}+\cdots+a_{20})=$ $\qquad$
Answer: 476837158203126
Problem: 设 $\alpha 、 \beta 、 \gamma$ 满足 $0<\alpha<\beta<\gamma<2 \pi$, 若对于任意 $x \in \mathbf{R}, \cos (x+\alpha)+\cos (x+\beta)+\cos (x+\gamma)=0$, 则 $\gamma-\alpha=$ ? 请给出答案的分子和分母之和。
Answer: 7
Problem: 若对一切 \(\theta \in \mathbf{R}\)，复数 \(z=(a+\cos \theta)+(2 a-\sin \theta) i\) 的模不超过 2 ，则实数 \(a\) 的取值范围为\([-p, q]\)。请给出 \(p + q\) 的值。
Answer: 0
Problem: 已知过点 $(0,1)$ 的直线 $l$ 与曲线 $C: y=x+\frac{1}{x}(x>0)$ 交于两个不同点 $M$ 与 $N$, 求曲线 $C$ 在点 $M 、 N$ 处切线的交点轨迹。请提供这两个交点坐标的所有整数部分之和。
Answer: 8
Problem: 已知 $p, q(q \neq 0)$ 是实数, 方程 $x^{2}-p x+q=0$ 有两个实根 $\alpha, \beta$,数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=p, a_{2}=p^{2}-q, \quad a_{n}=p a_{n-1}-q a_{n-2}(n=3,4, \cdots)$\n若 $p=1, q=\frac{1}{4}$, 求 $\left\{a_{n}\right\}$ 的前 $n$ 项和。请给出 $s_n$ 当 $n=2$ 时的值，并计算出结果为 $\frac{m}{n}$ 格式，给出 $m+n$ 的值。
Answer: 4
Problem: 已知 $\left\{a_{n}\right\}$ 是公差不为 0 的等差数列, $\left\{b_{n}\right\}$ 是等比数列, 其 中 $a_{1}=3, b_{1}=1, a_{2}=b_{2}, 3 a_{5}=b_{3}$, 且存在常数 $\alpha, \beta$ 使得对每一个正整数 $n$ 都有 $a_{n}=\log _{\alpha} b_{n}+\beta$, 则 $\alpha+\beta-\sqrt[3]{3}=$.
Answer: 3
Problem: 函数 $f(x)=a^{2 x}+3 a^{x}-2(a>0, a \neq 1)$ 在区间 $x \in[-1,1]$ 上的最大值为 8, 则它在这个区间上的最小值是 。已知原始答案的形式为$\frac{m}{n}$，请给出m + n的值。
Answer: 5
Problem: 正三棱柱 $A B C-A_{1} B_{1} C_{1}$ 的 9 条棱长都相等, $P$ 是 $C C_{1}$ 的中点, 二面角 $B-A_{1} P-B_{1} = \alpha$, 则 $\sin \alpha = \frac{\sqrt{m}}{n}$ 的形式，请给出 m + n 的值。
Answer: 14
Problem: 求所有三元整数组 $(x, y, z)$, 使其满足\n$$\n\left\{\begin{array}{l}\nx^{3}+y^{3}+z^{3}-3 x y z=2011, \\nx \geq 15, \\ny \geq 15.\n\end{array}\right.\n$$\n请提供满足条件的所有三元组的元素之和。
Answer: 5361
Problem: 已知 $x>0, y>0$, 且满足 \(\left\{\begin{array}{l}\cos ^{2}(\pi x)+2 \sin (\pi y)=1 &\quad①\\ \sin (\pi x)+\sin (\pi y)=0&\quad ②, \\ x^{2}-y^{2}=12&\quad③\end{array}\right.\)  
求有序数对 \((x, y)\) 中的 x + y 的值。
Answer: 6
Problem: 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 $x \in \mathbf{R}$, 都有 $f(x) \leqslant 0$, 求 $a$ 的取值范围;\n（2）若 $a \geqslant 2$ ，且存在 $x \in \mathbf{R}$ ，使得 $f(x) \leqslant 0$ ，求 $a$ 的取值范围。\n请提供所有区间端点的和。
Answer: 6
Problem: 复平面上，非零复数 $z_{1}, z_{2}$ 在以 $i$ 为圆心， 1 为半径的圆上， $\overline{z_{1}} z_{2}$ 的实部为零， $z_{1}$ 的辐角主值为 $\frac{\pi}{6}$ ，则 $z_{2}=$。请注意，$z_{2}$ 的原始答案为$-\frac{\sqrt{3}}{2}+\frac{3}{2}i$，请给出虚部分子和分母之和。
Answer: 5
Problem: 曲线 $C$ 的极坐标方程是 $\rho=1+\cos \theta$, 点 $A$ 的极坐标是 $(2,0)$, 曲线 $C$ 在它所在的平面内绕 $A$ 旋转一周，则它扫过的图形的面积是。原始答案为$\frac{m}{n}\pi$的形式，请给出m + n的值。
Answer: 19
Problem: 当 $\frac{\pi}{4} \leq x \leq \frac{\pi}{2}$ 时, $y=\sin ^{2} x+\sqrt{3} \sin x \cos x$ 的取值范围是 \qquad. 标准答案格式为[a,\frac{b}{c}]，请给出区间两端点a+b+c的和。
Answer: 6
Problem: 在 $\triangle A B C$ 中, $A B=1, A C=2, \cos B=2 \sin C$, 则 $B C$ 的长为。原始答案的格式为 $\frac{\sqrt{m}+\sqrt{n}}{k}$，请给出 m + n + k 的值。
Answer: 18
Problem: 数列 $\left\{a_{n}\right\}$ 满足: $a_{1}=a_{2}=a_{3}=1$ 。令 $$ b_{n}=a_{n}+a_{n+1}+a_{n+2}\left(n \in \mathbf{N}^{*}\right)。 $$ 若 $\left\{b_{n}\right\}$ 是公比为 3 的等比数列, 求 $a_{100}$ 的值。原始答案是 $a_{100}=\frac{m^{n}+a}{b}$的形式，请给出 m + n +a+b的值。
Answer: 126
Problem: 对任意三个两两不同的非负实数 $a, b, c$ ，定义
$$
S(a, b, c)=\frac{3 b+4 c}{|a-b|}+\frac{3 c+4 a}{|b-c|}+\frac{3 a+4 b}{|c-a|} ,
$$
并设 $S(a, b, c)$ 能取到的最小值为 $m_{0}$ .
求所有非负实数组 $(x, y, z)$ ，使得 $S(x, y, z)=m_{0}$ . 当t=1时，请给出这些数组中每个数组元素之和的总和。
Answer: 9
Problem: 在平面直角坐标系中, 圆 $\Omega$ 的方程为 $x^{2}+y^{2}-20 x-22 y-2022=0$, 则圆 $\Omega$ 的面积为。原始答案包含 $\pi$，请给出整数部分的值。
Answer: 2243
Problem: 设 \(\alpha, \beta \in \mathbf{R}\), 若 \(\tan (\alpha+\beta)=2, \tan (\alpha+2 \beta)=3\), 则 \(\tan \alpha\) 的值为\(\frac{m}{n}\)的格式，请给出\(m + n\)的值。
Answer: 22
Problem: 设实数 $k, l, m$ 满足：函数 $y=(x+1)\left(x^{2}+k x+l\right)$ 的图像有对称中心 $(1,0)$ ，且与函数 $y=x^{3}+m$ 的图像有公共点。已知 $k+l+m$ 的取值范围为一个区间。请给出区间上限的分子与分母之和。
Answer: 37
Problem: 若 \(\triangle A B C\) 满足 \(\frac{\angle A}{3}=\frac{\angle B}{4}=\frac{\angle C}{5}\), 则 \(\frac{\overrightarrow{A B} \cdot \overrightarrow{A C}}{|B C|^{2}}\) 的值为 \(\frac{m+\sqrt{n}}{k}\) 的形式，请给出 m + n + k 的值。
Answer: 10
Problem: 对任意正实数 $a$ ，记函数 $f(x)=|\lg x|$ 在 $[a,+\infty)$ 上的最小值为 $m_{a}$, 函数 $g(x)=\sin \frac{\pi x}{2}$ 在 $[0, a]$ 上的最大值为 $M_{a}$. 若 $M_{a}-m_{a}=\frac{1}{2}$, 求 $a$ 的所有可能值。已知原始答案形式为 \frac{m}{n} 或 \sqrt{m}，请给出 m + n 的最大值。
Answer: 10
Problem: 设实数 $a, b, c, d$ 满足 $a \geq b, c \geq d$ ，且  
$$  
|a|+2|b|+3|c|+4|d|=1.  
$$  
记 $P=(a-b)(b-c)(c-d)$ .求 $P$ 的最小值与最大值.  
原始答案是$\frac{m}{n}$的形式，请给出最小值和最大值的m + n的和。
Answer: 328
Problem: 给定实数 $r$ ，甲、乙两人玩如下的游戏. 黑板上写着一个含有三个绝对值的算式： $$ S=|\square-\square|+|\square-\square|+|\square-\square|, $$ 其中 6 个空格 "$\square$"中尚未填数.每一回合，甲选取区间 $[0,1]$ 中的一个实数（不同回合中可以选相同的数），乙将该数填在某个空格之中.经过六个回合之后所有 6 个空格中均填了数， $S$ 的值也随之确定.若 $S \geq r$ ，则甲胜，否则乙胜. 求所有的实数 $r$ ，使得甲有获胜策略. 原始答案为 $\frac{m}{n}$ 的格式，请给出 m + n 的值。
Answer: 23
Problem: 已知数列 $\left\{a_{n}\right\}$ 的各项均为非负实数, 且满足：对任意整数 $n \geq 2$, 均有 $a_{n+1}=a_{n}-a_{n-1}+n$. 若 $a_{2} a_{2022}=1$, 求 $a_{1}$ 的最大可能值。原始答案为\frac{m}{n}格式，请给出m + n的值。
Answer: 6076
Problem: 给定整数 $n(n \geq 2)$. 对于一个 $2 n$ 元有序数组 $$ T=\left(a_{1}, b_{1}, a_{2}, b_{2}, \cdots, a_{n}, b_{n}\right), $$ 若 $T$ 的每个分量均为 0 或 1 ，且对任意 $p, q(1 \leq p<q \leq n)$ ，均有 $\left(a_{p}, b_{p}, b_{q}\right) \neq(1,0,1)$ 且 $\left(a_{q}, b_{q}, a_{p}\right) \neq(1,0,0)$, 则称 $T$ 为 "有趣数组". 求有趣数组的个数。请提供当 n = 2 时的答案。
Answer: 12
Problem: 若正数 $a, b$ 满足 $\log _{2} a+\log _{4} b=8, \log _{4} a+\log _{8} b=2$, 则 $\log _{8} a+\log _{2} b$ 的值为。请给出原始答案中分子与分母绝对值之和的值。
Answer: 55
Problem: 设 $a$ 为实数. 若存在实数 $t$, 使得 $\frac{a-\mathrm{i}}{t-\mathrm{i}}+\mathrm{i}$ 为实数 ( $\mathrm{i}$为虚数单位), 则 $a$ 的取值范围是 $\qquad$. 求出4a的最大值。
Answer: -3
Problem: 设 $k, l, m$ 为实数, $m \neq 0$, 在平面直角坐标系中, 函数 $y=f(x)=k+\frac{m}{x-l}$的图像为曲线 $C_{1}$, 另一函数 $y=g(x)$ 的图像为曲线 $C_{2}$, 且满足 $C_{2}$ 与 $C_{1}$ 关于直线 $y=x$ 对称。若点 $(1,4),(2,3),(2,4)$ 都在曲线 $C_{1}$ 或 $C_{2}$ 上, 则 $f(k+l+m)$ 的值为整数，直接给出整数部分的值。
Answer: 1
Problem: 对于和为 1 的九个非负实数 $a_{1}, a_{2}, \cdots, a_{9}$, 令
$$
\begin{aligned}
& S=\min \left\{a_{1}, a_{2}\right\}+2 \min \left\{a_{2}, a_{3}\right\}+\cdots+8 \min \left\{a_{8}, a_{9}\right\}+9 \min \left\{a_{9}, a_{1}\right\}, \\
& T=\max \left\{a_{1}, a_{2}\right\}+2 \max \left\{a_{2}, a_{3}\right\}+\cdots+8 \max \left\{a_{8}, a_{9}\right\}+9 \max \left\{a_{9}, a_{1}\right\},
\end{aligned}
$$
这里, $\min \{x, y\}$ 表示 $x, y$ 中的较小者, $\max \{x, y\}$ 表示 $x, y$ 中的较大者.
记 $S$ 的最大可能值为 $S_{0}$ . 当 $S=S_{0}$ 时, 求 $T$ 的所有可能值. 计算区间两端分子分母之和的值。
Answer: 76