ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes, where each domino covers exactly two neighboring squares and the dominoes don’t overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$. Please provide the value of  k(C)  when  n = 11 . The answer should be in the format   \frac{m}{n}, directly give the value of  m + n .
Answer: 11
Problem: A random number selector can only select one of the nine integers $1, 2, ..., 9$, and it makes these selections with equal probability. Determine the probability that after $n$ selections ($n>1$), the product of the $n$ numbers selected will be divisible by $10$. Please calculate the probability when n = 2 and give the result in the form of \frac{m}{n}, and then provide the value of m + n.
Answer: 89
Problem: Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points? The original answer is in the format \frac{m}{k}, please provide the value of m + k when n = 1.
Answer: 4
Problem: On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points. The original answer is in \frac{m}{n} format, please give the value of m + n.
Answer: 3
Problem: 在$4n\times4n\left(n\in\mathbf{N}_{+}\right)$实数表中，每个数的绝对值均不超过1，所有数的和为0。已知数表中的每行各数之和的绝对值、每列各数之和的绝对值均不小于$c$。求$c$的最大可能值。请给出$c$的值，当$n=2$时。
Answer: 0
Problem: 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 $0 \leqslant x_{1} \leqslant x_{2} \leqslant \cdots \leqslant x_{n} \leqslant 1$ 的任意实数 $x_{i}(i=1,2, \cdots, n)$ ，均有
## $\min _{1 \leqslant i \leqslant n}\left|x_{i}-a_{i}\right| \leqslant \lambda$.
请提供当n = 1时，\lambda的值以分数形式\frac{m}{n}，并给出m + n的值。
Answer: 3
Problem: 求最大的实数 $c$ ，满足：对任意正整数 $n>1$ ，均存在 $\{1,2, \cdots, n\}$ 的子集 $A$ ，使得 $|A| \geqslant_{c n}$ ，且对任意的 $x 、 y_{、} z \in A$ （允许相同），均有 $n$ 不是 $x+y-z$ 的约数。请给出m + n的值，其中原始答案为\frac{m}{n}。
Answer: 4
Problem: Determine the smallest number $M$ such that the inequality
$$\begin{vmatrix}ab(a^2-b^2)+bc(b^2-c^2)+ca(c^2-a^2)\end{vmatrix}\le M\bigl(a^2+b^2+c^2\bigr)^2$$
holds for all real numbers $a,b,c.$ The original answer is in the format \frac{m}{n}\sqrt{2}, please give the value of m + n.
Answer: 11
Problem: Determine all pairs $(x,y)$ of integers satisfying the equation $1+2^x+2^{2x+1}=y^2$. Please provide the sum of all individual elements from each solution pair.
Answer: 0
Problem: Let $n>1$ be an integer. In the space, consider the set $S=\{(x,y,z)\mid x,y,z\in\{0,1,\cdots,n\}, x+y+z>0\}.$ Find the smallest number of planes that jointly contain all $(n+1)^3-1$ points of $S$ but none of them passes through the origin. Provide the value of this expression when n = 2.
Answer: 6
Problem: Find all pairs $(k,n)$ of positive integers for which $7^k-3^n$ divides $k^4+n^2$. Please provide the sum of all values of $k$ and $n$ in the pairs.
Answer: 4
Problem: Find all functions $f\colon(0,\infty)\to(0,\infty)$ such that $\frac{f(p)^2+f(q)^2}{f(r^2)+f(s^2)}=\frac{p^2+q^2}{r^2+s^2}$ for all $p,q, r, s > 0$ with $pq = rs$. There are multiple solutions for $f(x)$, represented by their respective function numbers. Please provide the sum of these numbers.
Answer: 0
Problem: Let $n$ and $k$ be fixed positive integers of the same parity, $k \ge n$. We are given $2n$ lamps numbered $1$ through $2n$; each of them can be on or off. At the beginning all lamps are off. We consider sequences of $k$ steps. At each step one of the lamps is switched (from off to on or from on to off). Let $N$ be the number of $k$-step sequences ending in the state: lamps $1,....,n$ on, lamps $n+1,...,2n$ off. Let $M$ be the number of $k$-step sequences leading to the same state and not touching lamps $n+1,....2n$ at all. Find the ratio $N/M$ when $k = n + 1$. Please provide the integer value of the ratio.
Answer: 1
Problem: Find all positive integers n such that there exists a sequence of positive integers $a_{1},a_{2},\\ldots,a_{n}$ satisfying $a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1$ for every k with $2\leq k\leq n-1.$ When n = 4, a sequence exists with specific values for $a_1$, $a_2$, $a_3$, and $a_4$. Please provide the sum of these values.
Answer: 3
Problem: Determine all functions f : $\mathbb{R}\to\mathbb{R}$ such that the equality $f([x]y)=f(x)[f(y)].\quad(1)$ holds for all $x,y\in\mathbb{R}$. Here, by $[x]$ we denote the greatest integer not exceeding $x$. Please provide the sum of all possible integer values of the constant C.
Answer: 1
Problem: Denote by $\mathbb{Q}^{+}$ the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^{+}\to\mathbb{Q}^{+}$ which satisfy the following equation for all $x,y\in\mathbb{Q}^{+}:$ $f\big(f(x)^2y\big)=x^3f(xy).\quad(1)$ The original answer is in \frac{m}{n} format for a specific x, please provide the value of m + n when x = 1.
Answer: 2
Problem: On some planet, there are $2^N$ countries ($N \ge 4)$. Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1$, each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set. Assume $N = 4$, provide the value of $M$.
Answer: 17
Problem: Find all pairs $( m, n)$ of nonnegative integers for which $$m^2+2\cdot3^n=m\big(2^{n+1}-1\big).\quad(1)$$ Please provide the sum of all values of $m + n$ for each pair $(m, n)$.
Answer: 21
Problem: For 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 $p_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i + a_j$ divides $s_A$. Determine the maximum value of $p_A$ among all sets of four distinct positive integers.
Answer: 4
Problem: Determine all sequences $(x_1,x_2,\cdots,x_{2011})$ of positive integers such that for every positive integer $n$ there is an integer $a$ with $$x_1^n+2x_2^n+\cdots+2011x_{2011}^n=a^{n+1}+1.$$ Please provide the value of $x_1 + x_2 + \cdots + x_{2011}$ for the sequence that satisfies the condition.
Answer: 2011
Problem: Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy $f^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1$ for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\cdots f}_k(n)\cdots)).$ Provide the value of $f(1) + g(1)$.
Answer: 1
Problem: Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself such that for all integers $x$ and $y$, the difference $f(x) − f(y)$ divides $x^n − y^n$. The original answer involves parameters \varepsilon, d, and c. Provide the sum of all distinct possible values of \varepsilon + d + c.
Answer: 0
Problem: For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the differences $\begin{aligned}t(n+a)-t(n), t(n+a+1)-t(n+1),\cdots,t(n+2a-1)-t(n+a-1)\end{aligned}$ are divisible by $4$. Please provide the sum of all possible values of $a$.
Answer: 9
Problem: Find 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 all integers $a,b,c$ satisfying $a+b+c=0.$ Please provide the sum of all elements within each tuple of possible values for f(a), f(b), and f(c).
Answer: 0
Problem: In a $999 × 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_1, C_2, C_3)$ of cells, the first two in the same row and the last two in the same column, with $C_1$ and $C_3$ white and $C_2$ red. Find the maximum value T can attain. The original answer is in \frac{m}{n} format, please give the value of m + n.
Answer: 3980000000000
Problem: Call admissible a set $A$ of integers that has the following property: If $x,y\in A$ (possibly $x=y)$ then $x^2+kxy+y^2\in A$ for every integer $k.$ Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. Please provide the sum of the pair $(m, n)$.
Answer: 1
Problem: Find 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 the sum of the values of x, y, and z for all such triples.
Answer: 505
Problem: Find all $n\in\mathbb{N}$ for which there exist nonnegative integers $a_1,a_2,\cdots,a_n$ such that $\frac{1}{2^{a_{1}}}+\frac{1}{2^{a_{2}}}+\cdots+\frac{1}{2^{a_{n}}}=\frac{1}{3^{a_{1}}}+\frac{2}{3^{a_{2}}}+\cdots+\frac{n}{3^{a_{n}}}=1$. Please provide the sum of all such possible values of $n$.
Answer: 37
Problem: Find the least positive integer $n$ for which there exists a set $\{s_1,s_2,\ldots,s_n\}$ consisting of $n$ distinct positive integers such that
$\left(1-\dfrac{1}{s_1}\right)\left(1-\dfrac{1}{s_2}\right)\dots\left(1-\dfrac{1}{s_n}\right)=\dfrac{51}{2010}.$
Please provide the value of $n$.
Answer: 39
Problem: In the plane, $2013$ red points and $2014$ blue points are marked so that no three of the marked points are collinear. One needs to draw $k$ lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors. Find the minimal value of $k$ such that the goal is attainable for every possible configuration of $4027$ points. Please provide the value of k.
Answer: 2013
Problem: $A \pm1-sequence$ is a sequence of 2022 numbers $a_{1},\ldots,a_{2022}$, each equal to either +1 or -1. Determine the largest $C$ so that, for any $\pm1$ sequence, there exists an integer $k$ and indices $1\leqslant t_{1}<\ldots<t_{k}\leqslant2022$ so that $t_{i+1}-t_{i}\leqslant2$ for all $i$, and $$\left|\sum_{i=1}^ka_{t_i}\right|\geqslant C.$$ Please provide the value of C.
Answer: 506
Problem: Solve in integers the equation $xy+3x-5y=-3.$ Please provide the value of the number of solutions multiplied by 2.
Answer: 24
Problem: Find integers $x, y, z, u$ such that $$x^2 + y^2 + z^2 + u^2 = 2xyzu.$$ Provide the sum of all integers $x, y, z, u$ that satisfy the equation.
Answer: 0
Problem: Determine 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 all integers $m$ and $n$. The original answer is in the form of a function $n\longmapsto2n+1007$. Please provide the value of the function when $n = 1000$. Directly give the resulting integer value.
Answer: 3007
Problem: Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has $|y^2-P(x)|\leqslant2|x|$ if and only if $|x^2-P(y)|\leqslant2|y|$. Determines the positive integer value in the set of possible values for P(0).
Answer: 1
Problem: Find all functions $f: \mathbb{Z} \to \mathbb{Z}$ such that $$n^2+4f(n)=f(f(n))^2\quad(1)$$ for all $n \in \mathbb{Z}.$ Provide the number of distinct functional forms for $f(n)$.
Answer: 3
Problem: Let $n\geqslant2$ be an integer. Consider an $n \times n$ chessboard divided into $n^ 2$ unit squares. We call a configuration of n rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that for every happy configuration of rooks, we can find a $k \times k$ square without a rook on any of its $k ^2$ unit squares. Please provide the value of $k$ when $n = 10$. The answer should be a single integer.
Answer: 3
Problem: $A$ card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of his cards so that each of the ten digits occurs on an even number of his remaining cards. If one player can do this but the other one cannot, the one who can is the winner; otherwise a draw is declared. Determine the number of possible first moves of the first player after which he has a winning strategy.What is the total number of possible moves
Answer: 1023
Problem: Let $n\geqslant 2$ be an integer, and let $A_n$ be the set
$$A_n=\{2^n-2^k\mid k\in\mathbb{Z},0\leqslant k<n\}.$$ Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n.$ Please provide the value when n = 9.
Answer: 3585
Problem: Find all triples $( p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^p-1+y$ and $x+y^p-1$ are both powers of $p.$ Provide the sum of the elements in all triples when k=1.
Answer: 24
Problem: Determine all functions $f:\mathbb{Z}\to\mathbb{Z}$ with the property that $$f\big(x-f(y)\big)=f\big(f(x)\big)-f(y)-1\quad(1)$$ holds for all $x,y\in\mathbb{Z}.$ Please provide several such functions.
Answer: 2
Problem: Let $n$ be a fixed positive integer. Find the maximum possible value of $$\sum\limits_{1\leqslant r<s\leqslant2n}(s-r-n)x_rx_s,$$ where $-1\leqslant x_{i}\leqslant1$ for all $i=1,2,\cdots,2n.$ Please provide the value of the expression when $n$ is a specific integer, such as $n=5$. The answer should be given in integer form, which is the product of n and (n-1).
Answer: 20
Problem: 已知非负实数 $a, b, c$ 满足 $a+b+c=1$, 则 $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)$ 的最大值为 \(\frac{m+\sqrt{n}}{k}\) 的形式，请给出 m + n + k 的值。
Answer: 12
Problem: 有限项等差数列的公差为 4 ，第二项起各项之和加上首项的平方小于 100 ，则该数列可能有几项？请提供可能项数的最小值与最大值之和。
Answer: 9
Problem: 有限项等差数列的公差为 4 ，第二项起各项之和+首项的平方 <100 ，则该数列可能有几项？请给出可能项数的最大值与最小值之和。
Answer: 9
Problem: $x_{1}, x_{2}, x_{3}, x_{4}$ 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, $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left\{x_{i 3}, x_{i 4}\right\}, Y=\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right.\right.$, 求 x 大于 y 的概率。原始答案是\frac{m}{n}格式，请给出m+n的值。
Answer: 4
Problem: 定义 $x * y=\frac{x+y}{1+x y}$, 则 $(\cdots((2 * 3) * 4) \cdots) * 21=$. 原始答案是 \( \frac{m}{n} \) 格式，请给出 \( m + n \) 的值。
Answer: 231
Problem: 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。
Answer: 33
Problem: 已知 $x_{1}, x_{2}, x_{3}, x_{4}$ 是两两不等的实数， $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 是 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意一个排列，若记 $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left(x_{i 3}, x_{i 4}\right\}\right\}, Y=\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right\}$ ，则 $X>Y$ 的概率是原始答案为\frac{m}{n}的格式，请给出m + n的值。
Answer: 4
Problem: 若非负数 $a, b, c$ 满足 $a+b+c=1$ ，则 $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)$ 的最大值是\n\n题目原始答案的格式为\(\frac{\sqrt{m}}{n}\)，请给出\(m + n + k\)的值，其中\(k = 0\)。
Answer: 21
Problem: 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 $	riangle A B P$ 的最小面积。请给出最小面积的数值。
Answer: 4
Problem: 已知 $\operatorname{Re}(z) \geq 1$ ，求 $\left|\frac{1}{z}+1+i\right|_{\min }$ 的值. 原始答案是 \(\frac{\sqrt{m}-n}{k}\) 的形式，请给出 m + n + k 的值.
Answer: 16
Problem: 已知 $\mathrm{I}_{\mathrm{n}}=\left(1+\frac{\mathrm{i}}{2}\right)\left(1+\frac{\mathrm{i}}{4}\right)\left(1+\frac{\mathrm{i}}{16}\right) \cdots\left(1+\frac{\mathrm{i}}{2^{2 \mathrm{n}}}\right)$ ，求 $\left|\mathrm{I}_{\mathrm{n}}\right|$ 当 $n = 2$ 时，给出整数近似值。
Answer: 1
Problem: 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 $S_{\triangle A B P}$ 的最小值和 $P$ 的轨迹。请给出最小值。
Answer: 4
Problem: 椭圆 $\frac{x^{2}}{4}+\frac{y^{2}}{3}=\lambda, F$ 为左焦点, $A, B$ 为椭圆上两点且 $|F A|=5,|F B|=8$, 求直线 $A B$ 的斜率 $k$ 的范围。请给出该范围上下限绝对值之和的近似整数值
Answer: 4
Problem: 已知 $\operatorname{Re}(z) \geq 1$, 求 $\left|\frac{1}{z}+1+\mathrm{i}\right|_{\text {min }}$ 的值。原始答案为 $\frac{\sqrt{m}-n}{k}$ 的格式，请给出 m + n + k 的值。
Answer: 16
Problem: 已知 $a x^{3}+b x^{2}+x+1=0(a<0)$ 恰有两个零点, 求 $a+b$ 的取值范围。标准答案格式为$\left(-\infty, \frac{a}{b}\right)$，请给出区间端点a+b的和。
Answer: 5
Problem: 已知 $I_{n}=
\left(1+\frac{i}{2}\right)\left(1+\frac{i}{4}\right)\left(1+\frac{i}{16}\right) \cdots\left(1+\frac{i}{2^{2^{n}}}\right)$, 求 $\left|I_{n}\right|$.\n\n原始答案的格式为\sqrt{\frac{m}{n}\left(1-\frac{1}{2^{2^{n+2}}}\right)}, 请给出m + n的值。
Answer: 7
Problem: Evaluate the expression: $\tan \left(\arctan 2+\arctan \frac{2}{2^{2}}+\cdots+\arctan \frac{2}{12^{2}}\right)$. The original answer is in the form $-\frac{m}{n}$. Please provide the value of $m + n$.
Answer: 31
Problem: 将一枚均匀的硬币连续抛郑 $n$ 次, 以 $p_{n}$ 表示未出现连续 3 次正面的概率.\n(1) 求 $p_{1}, p_{2}, p_{3}, p_{4}$;\n(2) 探究数列 $\left\{p_{n}\right\}$ 的递推公式，并给出证明;\n(3) 讨论数列 $\left\{p_{n}\right\}$ 的单调性及其极限，并阐述该极限的概率意义。\n请给出(1)部分中$p_{3}$和$p_{4}$分子与分母之和的值。
Answer: 44
Problem: Given the functions $$ f(x)=\frac{x}{\ln x}, $$ and $g(x)=k(x-1)$ where $k \in \mathbf{R}$.
(1) Prove that for any $k \in \mathbf{R}$, the line $y=g(x)$ is not a tangent to the curve $y=f(x)$;
(2) If there exists $x \in[\mathrm{e}, \mathrm{e}^{2}]$, such that $$ f(x) \leqslant g(x)+\frac{1}{2}, $$ find the range of values for $k$. Please translate the following text into English.
And give the value of \(2k\) when \(k\) takes the minimum value.
Answer: 1
Problem: 已知函数$f(x)=2 \cos x(\cos x+\sqrt{3} \sin x)-1(x \in \mathbf{R})$.(1) 求函数 $f(x)$ 的单调递增区间;(2) 设点 $P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right), \cdots$, $P_{n}\left(x_{n}, y_{n}\right), \cdots$ 均在函数 $y=f(x)$ 的图像上, 且满足$x_{1}=\frac{\pi}{6}, x_{n+1}-x_{n}=\frac{\pi}{2}\left(n \in \mathbf{Z}_{+}\right)$.求 $y_{1}+y_{2}+\cdots+y_{2018}$ 的值。请直接给出 $y_{1}+y_{2}+\cdots+y_{2018}$ 的值。
Answer: 0
Problem: 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 $O_{1}$ 和球 $O_{2}$ 都外切。则 $r$ 的值为原始答案为 $n-\sqrt{k}$ 的形式，请给出n + k的值。
Answer: 10
Problem: 设函数$$f(x)=\cos 2 x+2 \sqrt{3} \sin x^{\circ} \cos x(x \in \mathbf{R})$$的最大值为 $M$, 最小正周期为 $T$ 。(1) 求 $M 、 T$ 的值, 并写出函数 $f(x)$ 的单调递增区间；（2）若 10 个互不相等的正数 $x_{i}(i=1,2$, $\cdots, 10)$ 满足 $f\left(x_{i}\right)=M$ ，且 $x_{i}<10 \pi$ ，请给出满足条件的\frac{m\pi}{n}形式的答案中m+n的值。
Answer: 143
Problem: 设二次函数 $f(x)$ 在区间 $[-1,4]$ 上的最大值为 12, 且关于 $x$ 的不等式 $f(x)<0$ 的解集为区间 $(0,5)$ 。(1) 求函数 $f(x)$ 的解析式;(2) 若对于任意的 $x \in \mathbf{R}$, 不等式$f(2-2 \cos x)<f(1-\cos x-m)$恒成立, 求实数 $m$ 的取值范围。请提供不等式解集的区间端点之和。
Answer: -4
Problem: 设 $0<\alpha<\pi, \pi<\beta<2 \pi$. 若对任意的 $x \in \mathbf{R}$ ，都有
$$
\cos (x+\alpha)+\sin (x+\beta)+\sqrt{2} \cos x=0
$$
恒成立，试求 $\alpha, \beta$ 的值。答案为 $\frac{m\pi}{n}$ 的形式，请给出 $m+n$ 的值。
Answer: 11
Problem: 一个含有底面的半球形容器内放置有三个两两外切的小球。若这三个小球的半径均为 1 ，且每个小球都与半球的底面和球面相切，则该半球的半径 $R=$。原始答案为\frac{3+\sqrt{21}}{3}，请给出m + n + k的值。
Answer: 27
Problem: 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。
Answer: 2
Problem: 满足
$\frac{1-\sin \theta+\cos \theta}{1-\sin \theta-\cos \theta}+\frac{1-\sin \theta-\cos \theta}{1-\sin \theta+\cos \theta}=2$
的最大负角 $\theta$ 的弧度数为，原始答案为$-\frac{m\Pi}{n}$，请给出m + n的值。
Answer: 1
Problem: 设斜率为 \frac{\sqrt{2}}{2} 的直线 l 与椭圆 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0) 交于不同的两点 P 、 Q. 若点 P 、 Q在 x 轴上的射影恰好为植圆的两个焦点，则该椭圆的离心率是。原始答案为 \frac{\sqrt{m}}{n} 的形式，请给出 m + n 的值。
Answer: 4
Problem: 在侧棱长和底面边长都是 4 的正四棱锥 $P-A B C D$ 的表面上与顶点 $P$ 的距离为 3的动点所形成的所有曲线段的长度之和为，请给出\pi系数的值。
Answer: 6
Problem: 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 $a+b\sqrt{c}$，请给出a+b+c的值。
Answer: 8
Problem: 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 个自然数，见表1.\n| $a$ | $b$ | $c$ | $d$ | $e$ | $f$ | $g$ | $h$ | $i$ | $j$ | $k$ | $l$ | $m$ |\n| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |\n| $n$ | $o$ | $p$ | $q$ | $r$ | $s$ | $t$ | $u$ | $v$ | $w$ | $x$ | $y$ | $z$ |\n| 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 |\n给出明码对应的序号 $x$ 和密码对应的序号 $y$ 的变换公式：\n$$\ny= \begin{cases}\frac{x+1}{2}, & x \text { 为奇数,且 } 1 \leqslant x \leqslant 26 ; \\ \frac{x}{2}+13, & x \text { 为偶数,且 } 1 \leqslant x \leqslant 26.\end{cases}\n$$\n利用它可将明码转换成密码，如 $5 \rightarrow$ $\frac{5+1}{2}=3$, 即 $e$ 变成 $c ; 8 \rightarrow \frac{8}{2}+13=17$, 即 $h$ 变成 $q$ 。\n按上述公式，若将某明码译成的密码是\n$s h x c$ ，请提供原始明码中每个字母对应数字的和。
Answer: 54
Problem: 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 $A 、 B$.
(1) 证明: 直线 $A B$ 过定点;
(2) 求 $\triangle P A B$ 面积 $S$ 的最小值，以及取得最小值时点 $P$ 的坐标。请提供点P的坐标之和。
Answer: 0
Problem: 设点 $P 、 Q$ 分别在直线 $3 x-y+5=0$ 和 $3 x-y-13=0$ 上运动, 线段 $P Q$ 的中点为 $M\left(x_{0}, y_{0}\right)$, 且 $x_{0}+y_{0} \geqslant 4$. 求 $\frac{y_{0}}{x_{0}}$ 的取值范围的端点之和.
Answer: 4
Problem: 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 $	riangle P O Q$ 面积 $S$ 的取值范围。请给出区间两端点之和的整数值。
Answer: 0
Problem: 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 $\left[a_{n}\right]$ 和 $\left\{a_{n}\right\}$ 分别表示实数 $a_{n}$ 的整数部分与小数部分), 求 $a_{2020}$ 的整数部分
Answer: 3
Problem: 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。
Answer: 9
Problem: 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 $\angle C$ 的大小是？请提供 $\angle C$ 的度数的2倍值。
Answer: 90
Problem: 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。
Answer: 5
Problem: 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 \( m + n + k \) 的值，其中原始答案为 \( \frac{m+n\sqrt{2}}{k} \)。
Answer: 10
Problem: 已知函数
$f(x)=x^{3}-m x^{2}-x+1(m \in \mathbf{R})$.
(1) 求函数 $f(x)$ 的单调区间;
(2) 若对一切实数 $x$, 有
$f^{\prime}(x) \geqslant|x|-\frac{7}{4}$
成立，请给出 $m$ 的取值范围的区间端点之和。
Answer: -1
Problem: 已知椭圆 $\Gamma: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的两顶点为 $A(-2,0), B(2,0)$, 离心率为 $\frac{1}{2}$. 点 $P($ 不同于 $A 、 B)$ 在椭圆 $\Gamma$ 上, 点 $D(-4,0)$, $\overrightarrow{D E}=\frac{3}{5} \overrightarrow{D P}$, 直线 $A P$ 与 $B E$ 交于点 $Q$. 求点 $Q$ 的轨迹方程。请给出方程形式为 $( x + m )^{2} + \frac{n}{k}y^{2} = 1$ 中 m + n + k 的值。
Answer: 8
Problem: 过抛物线 $y=x^{2}$ 上两点 $A\left(1, y_{1}\right)$ 、 $B\left(x_{2}, y_{2}\right)$ 分别作抛物线的切线 $l_{1} 、 l_{2}, l_{1}$ 与 $l_{2}$交于点 $C$. 若 $\overrightarrow{A C} \cdot \overrightarrow{B C}=0$, 则 $x_{2}$ 的原始答案为 $-\frac{m}{n}$ 的形式，请给出 $m + n$ 的值。
Answer: 5
Problem: 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 $A D$ 上, 则二面角 $C$ $A B-D$ 的平面角的正弦值大小为 $\frac{\sqrt{m}}{n}$ 的形式，请给出m + n的值。
Answer: 9
Problem: 设函数 $f(x)=x-\ln (a x+2 a+1)+2$. 若对于任意的 $x \geqslant-2$, 均有 $f(x) \geqslant 0$, 请给出实数 $a$ 的取值范围 $[p, q]$ 中 $p + q$ 的值。
Answer: 1
Problem: 设 $P(0, a)$ 是 $y$ 轴上异于原点的任意一点, 过点 $P$ 且平行于 $x$ 轴的直线与曲线 $y=$ $\frac{1}{a} \ln x$ 交于点 $Q$, 曲线 $y=\frac{1}{a} \ln x$ 在点 $Q$ 处的切线交 $y$ 轴于点 $R$. 则 $\triangle P Q R$ 的面积的最小值是 $\qquad$。原始答案为 \(\frac{\sqrt{m\mathbf{e}}}{n}\) 格式，请给出 $m + n$ 的值。
Answer: 4
Problem: 给定正整数 $n(n \geqslant 2)$. 已知 $2 n$个正实数 $a_{1}, a_{2}, \cdots, a_{2 n}$ 满足: $\sum_{k=1}^{n} a_{2 k-1} \sum_{k=1}^{n} a_{2 k}=\prod_{k=1}^{n} a_{2 k-1}+\prod_{k=1}^{n} a_{2 k}$. 求 $S=\sum_{k=1}^{2 n} \frac{a_{k}^{n-1}}{a_{k+1}}$ 的最小值，其中 $a_{2 n+1}=a_{1}$ 。请给出当 n = 2 时的 S 的最小值。
Answer: 8