ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: On a table lies a regular 100-gon whose vertices are numbered consecutively: $1, 2, \cdots , 100$. These numbers were then rewritten, in increasing order, according to the distance of the corresponding vertex from the front edge of the table. If vertices are at the same distance from the edge, the left number is written first, and then the right one. All possible sets of numbers corresponding to different positions of the 100-gon are written out. Calculate the sum of the numbers in the 13-th position from the left in all these sets. Please provide the value of the sum of these numbers multiplied by 2.
Answer: 20200
Problem: A circuit of several resistors connects clamps A and B. Each resistor has an input and an output clamp. What is the least number of resistors needed for a circuit design that will not be short or open if any 9 resistors between A and B break? Please provide the sum of the resistors needed for both principal circuit designs.
Answer: 200
Problem: Let x and y be unknown digits. The 200-digit number $89 252 525 . . . 2 525$ is multiplied by the number $\overline{444 x18 y27}$. It turns out that the 53-rd digit from the right of the product is 1, and the 54-th digit is 0. Please provide the value of x + y.
Answer: 10
Problem: The decimal expression of a 100-digit number consists of 1’s and 2’s. The Rule allows one to select arbitrarily 10 consecutive digits of which the first five may change places with the second five. Two numbers will be called similar if one can be obtained from the other one in several such operations. What greatest number of such 100-digit numbers can be selected so no two of them are similar? Please provide the integer value of the original answer.
Answer: 4084101
Problem: We have 1985 sets. Each of the sets has 45 elements, the union of any two sets has exactly 89 elements. How many elements has the union of all these 1985 sets? Please provide the final integer value of the total number of elements in the union.
Answer: 87341
Problem: 对任意的正整数 $n$, 记 $D_{n}$ 为 $n$ 的正约数全体， $f_{i}(n)(i=1,2)$ 为集合
$$
F_{i}(n)=\left\{a \in D_{n} \mid a \equiv i(\bmod 4)\right\}
$$
的元素个数. 求最小的正整数 $m$, 使得
$2 f_{1}(m)-f_{2}(m)=2017$. 请给出答案表达式中所有数字的和的计算结果。
Answer: 2023
Problem: There are given segments $AB, CD$ and a point O such that no three of the points $A, B, C, D, O$ are on one straight line. The endpoint of a segment is marked if the straight line passing through it and O does not intersect another segment. How many marked endpoints are there? Please provide the sum of all possible numbers of marked endpoints.
Answer: 6
Problem: Nick and Pete divide between themselves $2n + 1$ nuts, $n ≥ 2$, and each tries to get the greater share, naturally. According to The Rule there are three ways to divide the nuts. Each way consists of three steps and the 1-st and 2-nd steps are common for all three ways. 
1-st step: Pete divides all nuts into two piles, each containing no less than two nuts. 
2-nd step: Nick divides both piles into two, each new pile containing no less than one nut. 
3-rd step: Nick sticks to either of the following methods: 
a) Nick takes either the biggest and the smallest pile, or 
b) Nick takes both medium-sized piles, or 
c) Nick choses either the biggest and the smallest or the medium-sized piles, but pays Pete one nut for the choice. 
Find the most profitable and the least profitable of methods a) – c) for Nick to divide the nuts.Please provide the integer value of the best method
Answer: 1
Problem: Find all positive integers n such that $(n − 1)!$ is not divisible by $n^2$. When p is equal to 1,please provide the sum of all such integers.
Answer: 20
Problem: Denote by $d(N)$ the number of divisors of N (1 and N are also considered as divisors of N). Find all N such that $N/d(N) = p$ is a prime. Please provide the sum of all possible values of N.
Answer: 59
Problem: Number 4 has the following property: when divided by $q^2$, for any q, the remainder is less than $\frac {q^2}{2}$. Find all numbers with the same property and provide the sum of these numbers.
Answer: 5
Problem: How may 50 cities be connected by the least possible number of airlines so that from any city one could get to any other by changing airplanes not more than once (i.e., using two planes)? Please provide the number of airlines needed.
Answer: 49
Problem: Is there a quadrilateral $ABCD$ of area 1 such that for any point O inside it the area of at least one of the triangles $\triangle OAB, \triangle OBC, \triangle OCD,$ or $\triangle OAD$ is an irrational number? The original answer involves a trapezoid with bases m and $\sqrt[n]{p}$. Please provide the sum of m，n and p involved in the original example.
Answer: 6
Problem: Is there a real number h such that $[h\cdot1969^n]$ is not divisible by $[h\cdot1969^{n-1}]$ for any positive integer n? The original answer is in \frac{m^3}{n} format, please give the value of m + n.
Answer: 3937
Problem: Each pan of a balance has k weights, numbered from 1 to k. The left pan is heavier. It turns out that if we interchange the places of any two weights with the same number, then either the right pan becomes heavier or the two pans reach an equilibrium. For what k this is possible? Please provide the sum of all possible values of k.
Answer: 3
Problem: A closed non-selfintersecting broken line is drawn on an infinite chessboard along the sides of its squares. There are K black squares inside the broken line. What is the greatest area of the figure bounded by the broken line when $K=4$?
Answer: 17
Problem: In a soccer tournament, n teams take part. Every team plays with the other one only once. Find the greatest difference between the final scores of the team with neighboring final positions, and express this difference as a function of n. Please provide the value of this difference when n = 10.
Answer: 10
Problem: Solve in positive integers $3 \cdot 2^x + 1 = y^2$. Please provide the sum of all integers from the solutions in the form (x, y).
Answer: 21
Problem: Given a collection of stones. The mass of each stone is $≤ 2 kg$ and their total mass is equal to $100 kg$. We selected a set of stones whose total mass differs from $10 kg$ by the least possible for this set number d. What is the greatest value of d for every admissible collection of stones? The original answer is in $\frac{m}{n}$ format, please give the value of $m + n$.
Answer: 21
Problem: On an $m × n$ rectangular piece of graph paper there are several squares whose sides are on the vertical and horizontal lines of the paper. It is known that no two squares coincide and no square is situated inside another one. What is the maximal number of such squares? Please provide the value when $m=3,n=4$.
Answer: 12
Problem: Consider a triangle. The radius of the inscribed circle is equal to $\frac 43$ ; the lengths of the triangle’s heights are integers whose sum is equal to 13. Find the lengths of the triangle’s sides.
If the sum of the squares of all sides is in the form of $\frac{m}{n}$, find the value of $m + n$.
Answer: 1871
Problem: $$\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}=a+\sqrt[b]{c}$$
Find the value of $a+b+c$.
Answer: 10
Problem: Find all positive integers n for which both $\frac{1}{n}$ and $\frac{1}{n+1}$ are finite decimal fractions. Provide the sum of all such n values.
Answer: 5
Problem: Find the total number of pairs of integers $(x, y)$ that satisfy the equation $x ^2 = y ^2 + 2y + 13$.
Answer: 4
Problem: The numbers $1, 2, . . . , 1983$ stand at the vertices of a regular 1983-gon. Any of the axes of symmetry of the 1983-gon divides the numbers which do not stand at the vertices through which the axis passes (if any) into two sets: on either side of the axis. Let us call an arrangement of numbers good with respect to a given axis of symmetry if every number of one set is greater than the number symmetrical to it. How many such good arrangements exist with respect to any axis of symmetry? Please provide the number of such arrangements.
Answer: 1
Problem: Three circles of radii $3, 4, 5$ are externally tangent to one another. The common tangent to the first two circles is drawn through the point at which they are tangent to each other. Find the length of this tangent contained inside the circle of radius 5. The original answer is in $\frac{m\sqrt{3}}{n}$ format, please give the value of $m + n$.
Answer: 47
Problem: Given the equation $\begin{aligned}\frac{x-49}{50}+\frac{x-50}{49}=\frac{49}{x-50}+\frac{50}{x-49}\end{aligned}$, if the sum of all possible values of $x$ is $\frac{b}{a}$, find the value of $a+b$.
Answer: 14801
Problem: Suppose the "complexity" of a number is the minimum possible length of a sequence (if any) that begins with 0 and ends with that number, with each subsequent term either equal to half of the previous term or equal to the sum of the previous term and the previous term, which is equal to 1. (The length of the empty sequence is assumed to be 0.) Find the number with the largest "complexity" among all numbers of the form $\frac m {2^{50}}$, where $m = 1, 3, 5, . . . , 2^{50} − 1.$. If the number $\frac{a}{b}+\frac{c}{b}\cdot\left(\frac{1}{2}\right)^{50}$ has the largest "complexity", equal to 101, find the value of $a+b+c$.
Answer: 6
Problem: Solve the equation:
$$(1 + x + x^ 2 )(1 + x + . . . + x^{10}) = (1 + x + . . . + x^6 ) ^2.$$
Please write the sum of all solutions for $x$.
Answer: -1
Problem: Given that in a convex quadrilateral $ABCD$, $\angle BAC=30^\circ, \angle ACD=40^\circ,\angle ADB=50^\circ,\angle CBD=60^\circ$ and $\angle ABC+\angle ADC>180^\circ.$, if $\angle A=\angle C=a^\circ, \angle B=b^\circ, \angle D=c^\circ.$, find the value of $a+b+c$.
Answer: 290
Problem: In the representation of numbers A and B as decimal fractions, the lengths of their minimal periods are equal to 6 and 12, respectively. What might the length of the minimal period in the similar representation of $A + B$ be? Find all answers and provide the sum of these possible lengths.
Answer: 16
Problem: Given n points on a plane, no three of them lie on a straight line. There is a straight line through each pair of points. When $n=5$, what is the minimum number of pairs of non-parallel lines among these lines?
Answer: 5
Problem: A fly is flying inside a regular tetrahedron with side a. The shortest distance for the fly to fly over every face and return to the initial position is $\frac{a}{\sqrt{m}}$. Find the value of $m$.
Answer: 10
Problem: In a round goblet whose section is the graph of the function $y = x^4$, a cherry — a ball of radius r — is dropped. What is the largest r for which the ball can touch the lowest point of the bottom? (In plain math words: what is the maximal radius of the disc lying in the domain $y ≥ x^4$ and containing the origin?) The original answer is in the format $\frac{m}{n}\sqrt[3]{2}$, please give the value of $m + n$.
Answer: 7
Problem: Let us refer to a rectangular of size$1 × k$ for any natural k a strip.For $n <= a$ and for $n > 2 · b + 1$can cut a $1995 × n$ rectangle into pairwise different strips,  Find the value of $a+b$.
Answer: 2993
Problem: 已知 $\triangle A B C$ 及其边 $B C$ 上的一点 $D$ 满足： $A B=2 B D$ ， $A C=3 C D$ ，且以 $A 、 D$ 为焦点可以作一个椭圆 $\Gamma$ 同时经过 $B 、 C$ 两点，求 $\Gamma$ 的离心率的$\sqrt{21}$倍。
Answer: 3
Problem: 已知抛物线 $y^{2}=2 p x$ 及定点 $A(a, b), B(-a, 0),\left(a b \neq 0, b^{2} \neq 2 p a\right). M$ 是抛物线上的点，设直线 $A M, B M$ 与抛物线的另一交点分别为 $M_{1}, M_{2}$ .  
求证：当 $M$ 点在抛物线上变动时（只要 $M_{1}, M_{2}$ 存在且 $M_{1} \neq M_{2}$ ）直线 $M_{1} M_{2}$ 恒过一个定点.那么当a=3时，求出这个定点的横坐标.
Answer: 3
Problem: 给定正整数 $n=3$. 求最大的实数 $\lambda$, 使得不等式 $a_{n}^{2} \geqslant \lambda\left(a_{1}+a_{2}+\cdots+a_{n-1}\right)+2 a_{n}$ 对任何满足 $a_{1}<a_{2}<\cdots<a_{n}$ 的正整数 $a_{1}, a_{2}, \cdots, a_{n}$ 均成立.
Answer: 1
Problem: 已知复数数列 $\left\{z_{n}\right\}$ 满足 $z_{1}=1, z_{n+1}=\overline{z_{n}}+1+n \mathrm{i}(n=1,2,3 \cdots)$ ，其中 $i$ 为虚数单位， $\overline{z_{n}}$ 表示 $z_{n}$ 的共轨复数，则 $z_{2015}$ 的实部为 $\qquad$.
Answer: 2015
Problem: 设实数 $a$ 满足 $a<9 a^{3}-11 a<|a|$, 则 $a$ 的最小值的$\sqrt{3}$是 $\qquad$.
Answer: -2
Problem: 函数 $y=x+\sqrt{x^{2}-3 x+2}$ 的最小值
Answer: 1
Problem: 设函数 $f(x)$ 的定义域为 $\mathbf{R}$, 且当 $x \geq 0$ 时, $f(x)=|x-2|+a$ (其中 $a$ 为实数). 若 $f(x)$ 为奇函数, 则不等式 $f(x) \geq 1$ 的最小值
Answer: -3
Problem: 设集合 $S$ 是 $\{0,1, \cdots, 98\}$ 的 $m(m \geqslant 3)$元子集，满足对任意的 $x, y \in S$ ，均存在 $z \in$ $S$, 使得 $x+y \equiv 2 z(\bmod 99)$. 求 $m$ 的所有可能值。请给出所有可能值的和。
Answer: 155
Problem: 已知 $f(x)$ 是定义在 $(0,+
fty)$ 上的减函数。若 $f\left(2 a^{2}+a+1\right)<f\left(3 a^{2}-4 a+1\right)$成立，请给出 $a$ 的取值范围端点之和的整数部分。
Answer: 6
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 $a$ 的取值范围的端点值之和。
Answer: 0
Problem: 在椭圆 $\Gamma$ 中, $A$ 为长轴的一个端点, $B$ 为短轴的一个端点, $F_{1}, F_{2}$ 为两个焦点. 若 $\overrightarrow{A F_{1}} \cdot \overrightarrow{A F_{2}}+\overrightarrow{B F_{1}} \cdot \overrightarrow{B F_{2}}=0$, 求 $5\tan \angle A B F_{1} \cdot \tan \angle A B F_{2}$ 的值。
Answer: -1
Problem: 不等式 $|x|^{3}-2 x^{2}-4|x|+3<0$ 的解集是 $x\in\left(a,b\right)\bigcup\left(c,d\right)$请问$a+d=\qquad$.
Answer: 0
Problem: 已知复数 $Z_{1} 、 Z_{2}$ 满足 $\left|Z_{1}\right|=2,\left|Z_{2}\right|=3$, 若它们所对应的向量的夹角为 $60^{\circ}$, 则 $7\sqrt{133}\left|\frac{Z_{1}+Z_{2}}{Z_{1}-Z_{2}}\right|=$ $\qquad$
Answer: 133
Problem: 设 $f(x)$ 是定义在 $\mathbf{R}$ 上的奇函数, 且当 $x \geqslant 0$ 时, $f(x)=x^{2}$ 。若对任意的 $x \in[a, a+2]$ ，不等式 $f(x+a) \geqslant 2 f(x)$ 恒成立，则实数 $a$ 的最小值的$\sqrt{2}$倍是
$\qquad$.
Answer: 2
Problem: 已知一个正三棱柱的各条棱长均为 $3$, 则其外接球的体积的$\frac{2 \sqrt{21}}{\pi}$为 $\qquad$.
Answer: 147
Problem: 记 [ $x$ ] 表示不超过实数 $x$ 的最大整数. 设 $x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{R}$, 且 $\left[x_{1}\right],\left[x_{2}\right], \cdots,\left[x_{n}\right]$ 为 $1, 2, \cdots, n$ 的一个排列, 其中, $n \geqslant 2$ 为给定整数. 当$n=6$时， 求 $\sum\limits_{i=1}^{n-1}\left[x_{i+1}-x_{i}\right]$ 的最大值
Answer: 5
Problem: 对有限非空实数集 $X$, 记 $X$ 的元素个数为 $|X|$ ，均有  
$f(X)=\frac{1}{|X|} \sum\limits_{a \in X} a$.  
集合对 $(A, B)$ 满足  
$A \cup B=\{1,2, \cdots, 100\}, A \cap B=\varnothing$,  
且 $1 \leqslant|A| \leqslant 98$.  
任取 $p \in B$ ，令  
$A_{p}=A \cup\{p\}, B_{p}=\{x \mid x \in B, x \neq p\}$.  
对所有满足上述条件的集合对 $(A, B)$ 与 $p \in$ $B$ ，求 $\left(f\left(A_{p}\right)-f(A)\right)\left(f\left(B_{p}\right)-f(B)\right)$ 的最大值时的$\frac{a}{b}$中的a+b.
Answer: 821
Problem: 已知  
$f(x)=\lg (x+1)-\frac{1}{2} \log _{3} x$.  
(1) 解方程: $f(x)=0$;
Answer: 9
Problem: 已知 $a_{1}, a_{2}, \cdots, a_{10}$ 和 $b_{1}, b_{2}, \cdots, b_{10}$ 均为 $1,2, \cdots, 10$ 的排列，即集合
$$
\begin{aligned}
& \left\{a_{1}, a_{2}, \cdots, a_{10}\right\} \\
& =\left\{b_{1}, b_{2}, \cdots, b_{10}\right\} \\
& =\{1,2, \cdots, 10\}.
\end{aligned}
$$
记 $c_{k}=a_{k}^{2}+b_{k}^{2}(k=1,2, \cdots, 10)$. 设 $c_{1}$, $c_{2}, \cdots, c_{10}$ 中的最大数为 $M$, 最小数为 $m$. 求
 $M$ 的最小可能值;
Answer: 101
Problem: 记 $[x]$ 表示不超过实数 $x$ 的最大整数。已知正整数数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=a$ ，且对任意正整数 $n$ ，均有 $$ a_{n+1}=a_{n}+2\left[\sqrt{a_{n}}\right] $$ (1) 若 $a=8$, 求最小的正整数 $n$, 使得 $a_{n}$为完全平方数； (2) 若 $a=2017$, 求最小的正整数 $n$, 使得 $a_{n}$ 为完全平方数。 请提供两个部分答案的和。
Answer: 87
Problem: $\sum\limits_{k=0}^{2022} C_{2022}^{k} \cos \frac{(1011-k) \pi}{2}$ 的值为$2^a$ ，$a=\qquad$.
Answer: 1011
Problem: 已知集合 $S=\{(-1,0),(1,0),(0,1),(0,-1)\}$, 甲虫第一天在原点 $O(0,0)$, 第 $n+1$ 天从第 $n$天的位置出发沿向量 $\frac{1}{4^{n}} \boldsymbol{v}$ 移动, 其中 $\boldsymbol{v} \in S$, 用 $S_{n}$ 表示第 $n$ 天甲虫可能在多少个不同的位置上，则 $S_{2023}=4^a$ ，请问$a=\qquad$
Answer: 2022
Problem: Determine the maximal length $L$ of a sequence $a_{1}, \\ldots, a_{L}$ of positive integers satisfying both the following properties: - every term in the sequence is less than or equal to $2^{2023}$, and - there does not exist a consecutive subsequence $a_{i}, a_{i+1}, \\ldots, a_{j}$ (where $1 \\leqslant i \\leqslant j \\leqslant L$ ) with a choice of signs $s_{i}, s_{i+1}, \\ldots, s_{j} \\in\\{1,-1\\}$ for which $$s_{i} a_{i}+s_{i+1} a_{i+1}+\\cdots+s_{j} a_{j}=0$$ Please provide the sum of the base and exponent used in the expression for the maximal length $L$.
Answer: 2026
Problem: Let $n$ be a positive integer. We arrange $1+2+\cdots+n$ circles in a triangle with $n$ rows, such that the $i^{\text {th }}$ row contains exactly $i$ circles. The following figure shows the case $n=6$. In this triangle, a ninja-path is a sequence of circles obtained by repeatedly going from a circle to one of the two circles directly below it. In terms of $n$, find the largest value of $k$ such that if one circle from every row is coloured red, we can always find a ninja-path in which at least $k$ of the circles are red. Please provide the smallest value of $n$ for which $k = 11$.
Answer: 1024
Problem: Determine all pairs $(a, p)$ of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all elements in these pairs.
Answer: 30
Problem: 给定正整数 $n(n \geqslant 2)$. 求最大的实数 $\lambda$, 使得不等式 $a_{n}^{2} \geqslant \lambda\left(a_{1}+a_{2}+\cdots+a_{n-1}\right)+2 a_{n}$ 对任何满足 $a_{1}<a_{2}<\cdots<a_{n}$ 的正整数 $a_{1}, a_{2}, \cdots, a_{n}$ 均成立。原始答案是 \frac{2n - 4}{n - 1} 的形式，请给出 m + n 的值，当 n = 3 时。
Answer: 1
Problem: 如果存在 $1,2, \cdots, n$ 的一个排列 $a_{1}, a_{2}, \cdots$, $a_{n}$, 使得 $k+a_{k}(k=1,2, \cdots, n)$ 都是完全平方数，则称 $n$ 为'好数'. 在集合 $\{11,13,15,17,19\}$ 中，哪些是'好数'，哪些不是'好数'？请提供'好数'和'不是好数'的个数之和。
Answer: 5
Problem: 设 $a 、 b 、 c$ 为正实数. 求
$\frac{a+3 c}{a+2 b+c}+\frac{4 b}{a+b+2 c}-\frac{8 c}{a+b+3 c}$
的最小值。标准答案格式为m+n\sqrt{k}的形式，给出结果为m+n+k的值。
Answer: -3
Problem: 求方程组
$$
\left\{\begin{array}{l}
5\left(x+\frac{1}{x}\right)=12\left(y+\frac{1}{y}\right)=13\left(z+\frac{1}{z}\right) \\
x y+y z+z x=1
\end{array}\right.
$$
的所有实数解。请提供所有解中各个变量的分数之和。
Answer: 0
Problem: 求出所有的正实数 $a$, 使得存在正整数 $n$ 个互不相交的无限整数集合 $A_1, A_2, \cdots, A_n$ 满足 $A_1 \cup A_2 \cup \cdots \cup A_n = \mathbb{Z}$, 而且对于每个 $A_i$ 中的任意两数 $b > c$, 都有 $b-c \geq a^i$。请给出该范围内的正整数之和的值。
Answer: 1
Problem: 给定实数 $a 、 b(a>b>0)$ ，将长为 $a$ 、宽为 $b$ 的矩形放入一个正方形内（包含边界)。问正方形的边至少为多长? 若 $a \geqslant(\sqrt{2}+1) b$, 则原始答案的形式为$\frac{\sqrt{m}}{n}(a+b)$，请给出m+n的值。
Answer: 4
Problem: $n$ 个棋手参加象棋比赛，每两个棋手比赛一局. 规定：胜者得 1 分，负者得 0 分，平局各得 0.5 分。如果赛后发现任何 $m$ 个棋手中都有一个棋手胜了其余 $m-1$ 个棋手，也有一个棋手输给了其余 $m-1$ 个棋手，就称此赛况具有性质 $P(m)$ 。对给定的 $m(m \geqslant 4)$ ，求 $n$ 的最小值 $f(m)$ ，使得对具有性质 $P(m)$ 的任何赛况，都有所有 $n$ 名棋手的得分各不相同。请提供当m=10时，f(m)的值。
Answer: 17
Problem: 求最小常数 $a(a>1)$, 使得对正方形 $A B C D$ 内部任一点 $P$, 都存在 $\triangle P A B 、 \triangle P B C$ 、 $\triangle P C D 、 \triangle P D A$ 中的某两个三角形，其面积之比属于区间 $\left[a^{-1}, a\right]$. 原始答案为 $\frac{a+\sqrt{b}}{c}$ 的形式，请给出a+b+c的值。
Answer: 8
Problem: 给定正整数 $m, n(2 \leq m \leq n)$, 设 $a_{1}, a_{2}, \cdots, a_{m}$ 是 $1,2, \cdots, n$ 中任取 $m$ 个互不相同的数构成的一个排列，如果存在 $k \in\{1,2, \cdots, m\}$ 使得 $a_{k}+k$ 为奇数，或者存在整数 $k, l(1 \leq k<l \leq m)$ ，使得 $a_{k}>a_{l}$ ，则称 $a_{1}, a_{2}, \cdots, a_{m}$ 是一个 "好排列"，试确定所有好排列的个数。当m=4, n=6时，请给出答案。
Answer: 355
Problem: 求最小常数 $a(a>1)$, 使得对正方形 $A B C D$ 内部任一点 $P$, 都存在 $\triangle P A B 、 \triangle P B C$ 、 $\triangle P C D 、 \triangle P D A$ 中的某两个三角形，其面积之比属于区间 $\left[a^{-1}, a\right]$. 请给出答案形式为 \frac{m + \sqrt{n}}{p}，并提供 m + n + p 的值。
Answer: 8
Problem: 已知一个有限图 G, 设 $f(\mathrm{G})$ 为图 G 中三角形的个数， $g(\mathrm{G})$ 为图 G 中的边数，求最小的常数 $c$ ，使得对于每个图 G, 均有 $f^{2}(\mathrm{G}) \leq c \cdot g^{3}(\mathrm{G})$. 已知原始答案的形式为 \frac{m}{n}, 请给出 m + n 的值。
Answer: 11
Problem: 在凸四边形 $A B C D$ 的外部分别作正 $\triangle A B Q 、 \triangle B C R 、 \triangle C D S 、 \triangle D A P$, 记四边形 $A B C D$ 的对角线的和为 $x$, 四边形 $P Q R S$ 的对边中点连线的和为 $y$. 求 $\frac{y}{x}$ 的最大值。原始答案为 $\frac{a+\sqrt{b}}{c}$ 的格式，请给出 a + b + c的值。
Answer: 6
Problem: 设正数列 $x_{1}, x_{2}, \cdots, x_{n}, \cdots$ 满足
$(8 x_{2}-7 x_{1}) x_{1}^{7}=8$
及 $x_{k+1} x_{k-1}-x_{k}^{2}=\frac{x_{k-1}^{8}-x_{k}^{8}}{(x_{k} x_{k-1})^{7}}(k \geqslant 2)$ 。
求正实数 $a$, 使得当 $x_{1}>a$ 时, 有单调性 $x_{1}>x_{2}>\cdots>x_{n}>\cdots$ ；当 $0<x_{1}<a$ 时，不具有单调性。标准答案格式为k^{\frac{m}{n}}，请给出k+m+n的值。
Answer: 17
Problem: 设正数列 $x_{1}, x_{2}, \cdots, x_{n}, \cdots$ 满足
$(8 x_{2}-7 x_{1}) x_{1}^{7}=8$
及 $x_{k+1} x_{k-1}-x_{k}^{2}=\frac{x_{k-1}^{8}-x_{k}^{8}}{(x_{k} x_{k-1})^{7}}(k \geqslant 2)$ 。
求正实数 $a$, 使得当 $x_{1}>a$ 时, 有单调性 $x_{1}>x_{2}>\cdots>x_{n}>\cdots$ ；当 $0<x_{1}<a$ 时，不具有单调性。请给出 $a$ 的值在 $\frac{m}{n}$ 格式下的 m + n 的值。
Answer: 9
Problem: 给定一个 $2008 \times 2008$ 的棋盘，棋盘上每个小方格的颜色均不相同。在棋盘的每一个小方格中填人 $C 、 G 、 M 、 O$ 这 4 个字母中的一个, 若棋盘中每一个 $2 \times 2$ 的小棋盘中都有 $C 、 G 、 M 、 O$ 这 4 个字母，则称这个棋盘为"和谐棋盘"。问有多少种不同的和谐棋盘？答案标准格式为$a \times b^{c}+d$，求出a+b+c+d的值。
Answer: 1998
Problem: 给定一个 $2008 \times 2008$ 的棋盘，棋盘上每个小方格的颜色均不相同。在棋盘的每一个小方格中填人 $C 、 G 、 M 、 O$ 这 4 个字母中的一个, 若棋盘中每一个 $2 \times 2$ 的小棋盘中都有 $C 、 G 、 M 、 O$ 这 4 个字母，则称这个棋盘为"和谐棋盘"。标准答案的格式为$a\times b^{c}-d$，计算出a+b+c+d的值。
Answer: 2046
Problem: 在平面直角坐标系中，设点集
$\{P_{1},P_{2},\cdots,P_{4n+1}\}$
$=\left\{(x,y)|x,y\text{ 为整数},|x|\leq n,|y|\leq n,xy=0\right\}$,
其中$n\in\mathbf{N}_+.$求
$$(P_{1}P_{2})^{2}+(P_{2}P_{3})^{2}+\cdots+(P_{n}P_{4n+1})^{2}+(P_{4n+1}P_{1})^{2}$$
的最小值。请提供当 $n=1$ 时的最小值。
Answer: 8
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。当 $n = 11$ 时，请给出答案。
Answer: 1024
Problem: 设集合 $S$ 是 $\{0,1, \cdots, 98\}$ 的 $m(m \geqslant 3)$元子集，满足对任意的 $x, y \in S$ ，均存在 $z \in$ $S$, 使得 $x+y \equiv 2 z(\bmod 99)$. 求 $m$ 的所有可能值。请给出所有可能值之和。
Answer: 155
Problem: 记 [ $x$ ] 表示不超过实数 $x$ 的最大整数. 设 $x_{1}, x_{2}, \\cdots, x_{n} \in \\mathbf{R}$, 且 $\left[x_{1}\right],\left[x_{2}\right], \\cdots,\left[x_{n}\right]$ 为 1, $2, \\cdots, n$ 的一个排列, 其中, $n \geqslant 2$ 为给定整数. 求 $\\sum_{i=1}^{n-1}\left[x_{i+1}-x_{i}\right]$ 的最大值的2倍和最小值的和。
Answer: 0
Problem: 给定整数 $n \geqslant 2$. 黑板上写着 $n$ 个集合，然后进行如下操作：选取黑板上两个互相不包含的集合 $A 、 B$ ，擦掉它们，然后写上 $A \cap B$ 与 $A \cup B$. 这称为一次操作. 如此操作下去，直到任意两个集合中均有一个包含另一个为止. 对所有的初始状态与操作方式, 当n=5时，求操作次数的最大可能值。
Answer: 10
Problem: 设 $Q$ 为有理数集, $Z$ 为整数集. 在坐标平面上，对正整数 $m$ ，定义点集 $$A_{m}=\left\{(x, y) \mid x, y \in \mathbf{Q}, x y \neq 0, \frac{x y}{m} \in \mathbf{Z}\right\}.$$ 对线段 $M N$, 定义 $f_{m}(M N)$ 为线段 $M N$ 上属于集合 $A_{m}$ 的点的个数. 求最小的实数 $\lambda$,使得对坐标平面上的任意直线 $l$, 均存在与 $l$有关的实数 $\beta(l)$, 满足: 对直线 $l$ 上任意两点 $M 、 N$,均有 $f_{2016}(M N) \leqslant \lambda f_{2015}(M N)+\beta(l)$. 原始答案是以 \frac{m}{n} 的形式，请给出 m + n 的值。
Answer: 2021
Problem: 给定素数 $p \geqslant 5$. 若 $p =7$，求三个连续正整数的乘积模 $p$ 的不同余数的个数。
Answer: 5
Problem: 求最大的实数 $C$ ，使得对任意的正整数 $n$ 和满足 $0=x_{0}<x_{1}<x_{2}<\cdots<x_{n}=1$ 的数列 $\left\{x_{k}\right\}$ ，均有$$\sum_{k=1}^{n} x_{k}^{2}\left(x_{k}-x_{k-1}\right)>C$$。原始答案是\frac{m}{n}的形式，请给出m+n的值。
Answer: 4
Problem: 设复数 $z=(\omega+2)^{3}(\omega-3)^{2}$ 。求当 $\omega$取遍所有模长为 1 的复数时, $z$ 的模长的最大值。原始答案为$m^{\frac{p}{q}}$的形式，其中$p,q$互质，请给出$m+p+q$的值。
Answer: 14
Problem: 给定正整数 $k$. 若整数集 $\mathbf{Z}$ 的子集序列 $I_{1} \supseteq I_{2} \supseteq \cdots \supseteq I_{k}$ 满足对 $i=1,2, \cdots, k, 168$ $\in I_{i}$ ，且对于任意的 $x 、 y \in I_{i}$ ，均有 $x-y \in I_{i}$ $\left(x 、 y\right.$ 可以相等），则称 $\left(I_{1}, I_{2}, \cdots, I_{k}\right)$ 为 $\mathbf{Z}$ 上的一条 $k$ —链。问： $\mathbf{Z}$ 上共有多少条 $k$ —链？请说明理由。请计算当k=5时的答案。请给出计算出的整数值。
Answer: 2016
Problem: 请给出正整数 $a_{1}, a_{2}, \cdots, a_{18}$, 使得 $a_{1}=1$, $a_{2}=2, a_{18}=2019$, 且对任意的 $k(3 \leqslant k \leqslant 18)$,均存在 $i 、 j(1 \leqslant i<j<k)$ 使得 $$ a_{k}=a_{i}+a_{j} $$ 请提供这些整数的和。
Answer: 5304
Problem: 给定整数 $n \geqslant 2$. 设 $x_{1}, x_{2}, \cdots, x_{n}$ 为任意实数. 求
$$
2 \sum_{1 \leqslant i<j \leqslant n}\left[x_{i} x_{j}\right]-(n-1) \sum_{i=1}^{n}\left[x_{i}^{2}\right]
$$
的最大值，其中， $[x]$ 表示不超过实数 $x$ 的最大整数。请给出当 n = 4 时的最大值。
Answer: 4
Problem: 求满足下面两个条件的所有实数序列 $\left\{b_{n}\right\} 、\left\{c_{n}\right\}(n \geqslant 1):$(1) 对于任意的正整数 $n$ ，均有 $b_{n} \leqslant c_{n}$ ；(2) 对于任意的正整数 $n$,均有 $b_{n+1} 、 c_{n+1}$为一元二次方程 $x^{2}+b_{n} x+c_{n}=0$ 的两根。若已知这两个数列是常数列，写出该常数值。
Answer: 0
Problem: 求所有的三元正整数组 $(a, b, c)$ 满足 $\frac{a}{2^{a}}=\frac{b}{2^{b}}+\frac{c}{2^{c}}$. 请提供满足条件的三元组元素和的总和。
Answer: 49
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.已知最终答案形式为$\frac{m}{n)^{2}$，其中$m,n$互质.求$m+n$.
Answer: 8388607
Problem: 求所有的正整数对 $(x, y)$ ，满足 $x^{y}=y^{x-y}$. 请给出所有满足条件的整数对的所有数字之和。
Answer: 24
Problem: 设集合 $S$ 是 $\{0,1, \cdots, 98\}$ 的 $m(m \geqslant 3)$元子集，满足对任意的 $x, y \in S$ ，均存在 $z \in$ $S$, 使得 $x+y \equiv 2 z(\bmod 99)$. 求 $m$ 的所有可能值的和.
Answer: 155