ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: In a $25 \times n$ grid, each square is colored with a color chosen among $8$ different colors. Let $n$ be as minimal as possible such that, independently from the coloration used, it is always possible to select $4$ columns and $4$ rows such that the $16$ squares of the intersections are all of the same color. Find the remainder when $n$ is divided by $1000$.
Answer: 601
Problem: Let $ABC$ be a triangle and $D$, $E$, and $F$ be the midpoints of sides $BC$, $AC$, and $BC$ respectively. The medians $AD$ and $BE$ are perpendicular, with $AD = 12$ and $BE = 9$. What is the value of $CF$?
Answer: 15
Problem: Positive numbers are written in the squares of a $10 \times 10$ table. Frogs sit in five squares and cover the numbers in these squares. Kostya found the sum of all visible numbers and got 10. Then each frog jumped to an adjacent square and Kostya’s sum changed to $10^2$. Then the frogs jumped again, and the sum changed to $10^3$ and so on: every new sum was 10 times greater than the previous one. What maximum sum can Kostya obtain?
Answer: 1000000
Problem: For each positive integer $k$, denote $C(k)$ as the sum of its distinct prime divisors. For example, $C(1) = 0$, $C(2) = 2$, and $C(45) = 8$. Find all positive integers $n$ for which $C(2^n + 1) = C(n)$.
Answer: 3
Problem: Find all real numbers $x$ such that $4x^5 - 7$ and $4x^{13} - 7$ are both perfect squares.
Answer: 2
Problem: Denote by $s(x)$ the sum of the digits of a positive integer x. Solve:
 x + s(x) + s(s(x)) + s(s(s(x))) = 1993$
Answer: 1963
Problem: 是否存在一个无穷正整数数列，使得对任意正整数 k，数列的任意连续 k 项之和均为 k+1 的倍数？
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 0
Problem: Javiera and Claudio play on a board consisting of a row with $2019$ cells. Claudio starts by placing a token anywhere on the board. Next, Javiera says a natural number $k$, where $1 \leq k \leq n$, and Claudio must move the token to the right or to the left by $k$ squares, at his choice, and so on. Javiera wins if she manages to remove the piece that Claudio moves from the board. Determine the smallest $n$ such that Javiera always wins after a finite number of moves.
Answer: 1010
Problem: Given that $x$, $y$ are positive integers such that $x(x+1)$ divides $y(y+1)$, but neither $x$ nor $x+1$ divides either $y$ or $y+1$. Find the smallest possible value of $x^2 + y^2$.
Answer: 596
Problem: Determine all possible values of the expression $xy + yz + zx$ given that $x, y, z$ are real numbers satisfying the conditions $x^2 - yz = y^2 - zx = z^2 - xy = 2$.
Answer: -2
Problem: Find all prime numbers $p$ such that $5^p + 4p^4$ is a perfect square.
Answer: 5
Problem: In a room, there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you turn on the $n$-th switch, the $n$-th bulb will light up. There is a group of men who are operating the switches according to the following rule: 

- They enter one by one and start flipping the switches starting from the first switch until they have to turn on a bulb. 
- As soon as they turn a bulb on, they leave the room.

For example:
- The first person enters, turns the first switch on, and leaves.
- The second person enters, sees the first switch on, turns it off, and then lights the second bulb.
- The third person enters, finds the first switch off, turns it on, and leaves the room.
- The fourth person enters, switches off the first and second bulbs, and switches on the third.

This process continues in this manner. Finally, we find out that the first 10 bulbs are off and the 11th bulb is on. How many people were involved in the entire process?
Answer: 1024
Problem: 设 p 是正实数. 已知直线 l 分别与两抛物线 C_1:y^2=2px 和 C_2:y^2=4px 交于四个不同的点 A(x_1,y_1), B(x_2,y_2), D(x_3,y_3), E(x_4,y_4)，且 y_4<y_2<y_1<y_3. 设 l 与 x 轴交于点 M，若 AD=6BE，求 \df{AM}{ME} 的值.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 3
Problem: The kid cut out a rectangle from grid paper, with each cell having a side length of 1, and calculated its area and perimeter. Carlson then took the scissors and cut out a square from this rectangle, along the grid lines, that was adjacent to the boundary of the rectangle.

- "My rectangle..." the kid sobbed. "There is something strange about this figure!"
- "Nonsense, do not mention it," Carlson said, waving his hand carelessly. "Here you see, in this figure, the perimeter is the same as the area of the rectangle was, and the area is the same as the perimeter was!"

What size square did Carlson cut out?
Answer: 2
Problem: Jay has a $24 \times 24$ grid of lights, all of which are initially off. Each of the $48$ rows and columns has a switch that toggles all the lights in that row and column, respectively, i.e., it switches lights that are on to off and lights that are off to on. Jay toggles each of the $48$ rows and columns exactly once, such that after each toggle, he waits for one minute before the next toggle. Each light uses no energy while off and 1 kiloJoule of energy per minute while on. To express his creativity, Jay chooses to toggle the rows and columns in a random order. Compute the expected value of the total amount of energy in kiloJoules expended by all the lights after all $48$ toggles.
Answer: 9408
Problem: 若一个五元整数组的元素可按某种顺序标记为 a, b, c, d, e，使得 a-b+c-d+e=29，则称此五元整数组为"可安置的". 求所有的 2017 元整数组 (n_1,n_2,\cdots,n_{2017})，满足：若将这 2017 个数依次按顺时针方向排列在圆周上（n_{2017} 与 n_1 相邻），则处于圆周上连续相邻位置的任一五元整数组均是可安置的.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 29
Problem: 一副三色纸牌共 32 张：其中红色、黄色、蓝色的牌各 10 张，编号分别为 1,2,\cdots,10；另有大小王牌各一张，编号均为 0. 现从这副牌中取若干张并计分，规定每张编号为 k 的牌计 2^k 分. 若一些牌的分值之和为 2004，则称这些牌为一个"好牌组". 求好牌组的个数.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 1006009
Problem: Consider a graph $G$ with 2019 nodes. Define "inverting" a vertex $v$ as follows: for every other vertex $u$, if there was an edge between $v$ and $u$, it is deleted; if there wasn't, it is added. We aim to minimize the number of edges in the graph through several inverting operations (we can invert the same vertex multiple times). Determine the smallest number $M$ such that we can always reduce the number of edges in the graph to be no greater than $M$, regardless of the initial configuration of $G$.
Answer: 1018081
Problem: The number $125$ can be expressed as a sum of pairwise coprime integers, each greater than $1$. Determine the maximum number of terms possible in such a sum.
Answer: 8
Problem: 设函数 f(x)=\sin x\cdot\sin3x，若关于 x 的方程 f(x)=a 在 (0,\pi] 上有奇数个不同的实数解，则实数 a 的值为__________.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: -1
Problem: Does there exist a polynomial of one variable that is irreducible over the field of integers, has degree $60$, and is a multiple of the form $X^n - 1$? If so, how many such polynomials exist?
Answer: 9
Problem: Find all positive integers $N$ that have at least 4 positive divisors such that the sum of the squares of the 4 smallest positive divisors of $N$ equals $N$.
Answer: 130
Problem: Determine the smallest natural number $n$ such that among any $n$ integers, one can choose $18$ integers whose sum is divisible by $18$.
Answer: 35
Problem: 已知 f(x)=\ln(1+x)(x>0)，定义函数序列 \{f_n(x)\}，其中 f_n 是 f 的 n 次迭代. 求所有的 x，使得 \displaystyle\lim_{n\rightarrow+\infty}nf_n(x) 存在，并求出该极限.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 2
Problem: A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times.  
 Sir Alex wants for each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum number of players required?
Answer: 18
Problem: Two linear functions $f(x)$ and $g(x)$ satisfy the following properties for all $x$:

- $f(x) + g(x) = 2$
- $f(f(x)) = g(g(x))$

Additionally, $f(0) = 2022$. Compute $f(1)$.
Answer: 1
Problem: The value of the expression $$\sum_{n=2}^{\infty} \frac{\binom{n}{2}}{7^{n-2}} = 1+\frac{3}{7}+\frac{6}{49}+\frac{10}{343}+\frac{15}{2401}+\dots+\frac{\binom{n}{2}}{7^{n-2}}+\cdots$$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 559
Problem: Find the sum of all integers $n$ satisfying the following inequality:

\[
\frac{1}{4}<\sin\frac{\pi}{n}<\frac{1}{3}.
\]
Answer: 33
Problem: $f(x)=\left\{\begin{array}{ll}x^{a} \sin \left(\frac{1}{x^{3}}\right) & x>0, a \in N_{+} \\ 0 & x \leq 0\end{array}\right.$,

$f(x)$ 连续可导且 $f^{\prime}(x)$ 处处连续, 求 $a_{\text {min }}$ 
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 5
Problem: 求三条直线 x+y=60, y=\frac{1}{2} x, y=0 所围成三角形上的整点个数
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 100
Problem: A natural number $n$ is called perfect if it is equal to the sum of all its natural divisors other than $n$. For example, the number $6$ is perfect because $6 = 1 + 2 + 3$. Find all even perfect numbers that can be expressed as the sum of two cubes of positive integers.
Answer: 28
Problem: 求不等式组\left\{\begin{array}{l}y<2 x \\ y>\frac{1}{2} x \\ x+y \leqslant 60\end{array}\right. 的整数解的组数.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 590
Problem: There are $10001$ students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold:

1. Each pair of students is in exactly one club.
2. For each student and each society, the student is in exactly one club of the society.
3. Each club has an odd number of students. In addition, a club with $2m+1$ students ($m$ is a positive integer) is in exactly $m$ societies.

Find all possible values of $k$.
Answer: 5000
Problem: If positive reals $x, y$ satisfy the equation $2(x+y) = 1 + xy$, find the minimum value of the expression $$A = x + \frac{1}{x} + y + \frac{1}{y}.$$
Answer: 8
Problem: Let $a$ and $b$ be real numbers such that $$\left(8^a+2^{b+7}\right)\left(2^{a+3}+8^{b-2}\right)=4^{a+b+2}.$$ The value of the product $ab$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 271
Problem: Let $S$ be the solid in three-dimensional space consisting of all points ( $x, y, z$ ) satisfying the following system of six simultaneous conditions:
$$
\begin{aligned}
& x \geqslant 0, \quad y \geqslant 0, \quad z \geqslant 0 \\
& x+y+z<11, \\
& 2 x+4 y+3 z \leqslant 36 \\
& 2 x+3 z \leqslant 24.
\end{aligned}
$$
 Determine the number $v$ of vertices of $S$.
Answer: 7
Problem: 独立掷 n 次硬币，每次只出现正面或反面. 用 a(n) 表示出现正面的次数恰好是 3 的倍数的情况总数，用 b(n) 表示出现正面的次数恰好是 6 的倍数的情况总数.(1) 求 a(2016),b(2016)；(2) 当 n\leq2016 时，求使得 2b(n)>a(n) 的正整数 n 的个数.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 840
Problem: 一项 "过关游戏"规则规定：在第 $n$ 关要抛掷一颗骰子 $n$ 次，如果这 $n$ 次抛掷所出现的点数的和大于 $2^{n}$ ，则算过关。问：
某人在这项游戏中最多能过几关?
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 4
Problem: Suppose $a$ and $b$ are real numbers such that the polynomial $x^3 + ax^2 + bx + 15$ has a factor of $x^2 - 2$. Find the value of $a^2b^2$.
Answer: 225
Problem: Let $S$ be the solid in three-dimensional space consisting of all points ( $x, y, z$ ) satisfying the following system of six simultaneous conditions:
$$
\begin{aligned}
& x \geqslant 0, \quad y \geqslant 0, \quad z \geqslant 0 \\
& x+y+z<11, \\
& 2 x+4 y+3 z \leqslant 36 \\
& 2 x+3 z \leqslant 24.
\end{aligned}
$$
Determine the number $e$ of edges of $S$.
Answer: 11
Problem: 设正整数 a,b 使得 15a+16b 和 16a-15b 都是正整数的平方. 求这两个平方数中较小者的最小值.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 231361
Problem: 1993年 , 美国 数学家  F. Smarandache 提出许多数 论 问 题 , 引起国内 外相关学者的关注 。 其 中之一便是著名的Smarandache 函数.正整数$n$的 Smarandache 函数定义为
$S(n)=\operatorname*{min}\{m|m\in\mathbf{N}_{+},n\mid m!\}$,比如：$S(2)=2,S(3)=3,S(6)=3.$
若$S(n)=7$,求正整数$n$的最大值
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 5040
Problem: Find all integers $n$ such that $n^{n-1} - 1$ is square-free.
Answer: 2
Problem: Determine all positive integers $n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y,z) = (x - y)^{5n} + (y - z)^{5n} + (z - x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients.
Answer: 1
Problem: Determine all natural numbers $n$ such that $9^n - 7$ can be represented as a product of at least two consecutive natural numbers.
Answer: 1
Problem: 已知 $a, b, c$ 是非负实数, 且 $S=a+2 b+3 c, T=a+b^{2}+c^{3}$.   
若 $S=4$, 求 $T$ 的最大值.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 4
Problem: 不等式 \log_{14}(\sqrt{x}+\sqrt[3]{x}+\sqrt[6]{x})^6\geq\log_2x 的解集是__________.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 64
Problem: On a circumference of a circle, seven points are selected, at which different positive integers are assigned to each of them. Then simultaneously, each number is replaced by the least common multiple of the two neighboring numbers to it. If the same number $n$ is obtained in each of the seven points, determine the smallest possible value for $n$.
Answer: 420
Problem: 在一个 (4n+2)\times(4n+2) 的方格表中，一只乌龟可以从一个格爬到与其相邻的另一个格中. 已知乌龟从某个角的格出发，经过每个格恰一次，最后回到出发的格. 求最大的正整数 k，使得乌龟总能进入某行或某列至少 k 次.
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 2
Problem: Let $f(x) = 5x^{13} + 13x^5 + 9ax$. Find the least positive integer $a$ such that $65$ divides $f(x)$ for every integer $x$.
Answer: 63
Problem: For what integers $n \geq 3$ is it possible to arrange the numbers $1, 2, \ldots, n$ in a circular order such that every number divides the sum of the next two numbers, in a clockwise direction?
Answer: 3
Problem: In a circle with radius $10$, point $M$ is located on the chord $PQ$ such that $PM = 5$ and $MQ = 10$. Through point $M$, we draw chords $AB$ and $CD$. Points $X$ and $Y$ are the intersection points of chords $AD$ and $BC$ with chord $PQ$, respectively. Given that $XM = 3$, find the length of $MY$.
Answer: 2
Problem: 已知 $F$ 为抛物线 $y^{2}=4 x$ 的焦点, 点 $M(4,0)$, 过点 $F$ 作斜率为 $k_{1}$ 的直线与抛物线交于点 $A 、 B$, 延长 $A M 、 B M$ 与抛物线分别交于点 $C 、 D$, 设直线 $C D$ 的斜率为 $k_{2}$.
求 $\frac{k_{1}}{k_{2}}$ 的值
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 4
Problem: Determine all integers $n > 1$ whose positive divisors add up to a power of $3$.
Answer: 2
Problem: The nonzero real numbers $a, b, c$ satisfy the following system of equations:
\[
\begin{cases} 
    a + ab = c \\
    b + bc = a \\
    c + ca = b 
\end{cases}
\]
Find all possible values of the product $abc$.
Answer: 3
Problem: A positive integer $n$ is considered "funny" if for every positive divisor $d$ of $n$, the number $d+2$ is a prime number. Determine all funny numbers that have the maximum possible number of divisors.
Answer: 135
Problem: 已知 $a, b, c$ 是非负实数, 且 $S=a+2 b+3 c, T=a+b^{2}+c^{3}$.  
 求 $T-S$ 的最小值
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: -3
Problem: Let $n$ be a positive integer such that no matter how $10^n$ is expressed as the product of two positive integers, at least one of these two integers contains the digit 0. Find the smallest possible value of $n$.
Answer: 8
Problem: In rhombus $ABCD$, let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. If $CN = 7$ and $DM = 24$, compute $AB^2$.
Answer: 250
Problem: Joana divided \(365\) by all integers from \(1\) to \(365\) and added all the remainders. Then she divided \(366\) by all the integers from \(1\) to \(366\) and also added all the remainders. Which of the two sums is greater and what is the difference between them?
Answer: 13
Problem: Let $a$ and $b$ be positive integers such that $2a - 9b + 18ab = 2018$. Find $b - a$.
Answer: 223
Problem: For a given positive integer $k$, denote the square of the sum of its digits by $f_1(k)$. Define the function recursively as $f_{n+1}(k) = f_1(f_n(k))$. Determine the value of $f_{1991}(2^{1990})$.
Answer: 256
Problem: Determine the minimum possible amount of distinct prime divisors of $19^{4n} + 4$, where $n$ is a positive integer.
Answer: 3
Problem: Let $p, q$ be prime numbers such that $n^{3pq} - n$ is a multiple of $3pq$ for all positive integers $n$. Find the least possible value of $p + q$.
Answer: 28
Problem: 是否存在两两不同的整数 a_1,a_2,\cdots，同时满足：(1) 对任意正整数 k，a_{k^2}>0 且 a_{k^2+k}<0；(2) 对任意正整数 n，|a_{n+1}-a_n|\leq2023\sqrt{n}？
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 1
Problem: Let $m, n, p$ be fixed positive real numbers which satisfy $m n p=8$. Depending on these constants, find the minimum of  
$$  
x^{2}+y^{2}+z^{2}+m x y+n x z+p y z,  
$$  
where $x, y, z$ are arbitrary positive real numbers satisfying $x y z=8$. When is the equality attained?  
Solve the problem for:   $m=n=p=2$
Answer: 36
Problem: 桌上有两个盒子，初始时，其中一个是空的，另一个有 29 个不同颜色的石子. 下面对这两个盒子轮流进行如下操作，从满的盒子开始. 在每次操作中，将当前盒子中的至少一个石子移动到另一个盒子中. 如果各次操作移动的石子不完全相同，问：至多能进行多少次操作？
让我们一步一步地思考。请以“Answer: \boxed{<final_answer>}”的格式输出最终答案。如果是选择题，请按顺序输出正确的选项，不带任何标点或空格。对于其他类型的问题，请只输出最终答案的数值。
Answer: 29
Problem: A set of positive integers is said to be *pilak* if it can be partitioned into two disjoint subsets $F$ and $T$, each with at least two elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers. 

Find all positive integers $n$ such that the set containing all the positive divisors of $n$ except $n$ itself is pilak.
Answer: 30
Problem: More than five competitors participated in a chess tournament. Each competitor played exactly once against each of the other competitors. Five of the competitors lost exactly two games each. All other competitors won exactly three games each. There were no draws in the tournament. Determine how many competitors there were and show a tournament that verifies all conditions.
Answer: 12
Problem: For any natural number, let $S(n)$ denote the sum of the digits of $n$. Find the number of 3-digit numbers for which $S(S(n)) = 2$.
Answer: 100
Problem: What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$?
Answer: 20
Problem: Let $a_n=6^{n}+8^{n}$ . Determine the remainder upon dividing $a_ {83}$ by $49$ .
Answer: 35
Problem: Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
Answer: 57
Problem: What is the largest $2$ -digit prime factor of the integer $n = {200\choose 100}$ ?
Answer: 61
Problem: The numbers $1447$ , $1005$ and $1231$ have something in common: each is a $4$ -digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there?
Answer: 432
Problem: Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$.

[asy] draw(circle((0,0),4)); draw((-4,0)--(4,0)); draw((-2,-2*sqrt(3))--(-2,2*sqrt(3))); draw((-2.6,0)--(-2.6,0.6)); draw((-2,0.6)--(-2.6,0.6)); dot((0,0)); dot((-2,0)); dot((4,0)); dot((-4,0)); dot((-2,2*sqrt(3))); dot((-2,-2*sqrt(3))); label("A",(-4,0),W); label("B",(4,0),E); label("C",(-2,2*sqrt(3)),NW); label("D",(-2,-2*sqrt(3)),SW); label("H",(-2,0),SE); label("O",(0,0),NE);[/asy]
Answer: 65
Problem: For $\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1, 2, 3, 6,9\}$ is $9-6+3-2+1=5$ and for $\{5\}$ it is simply $5$ . Find the sum of all such alternating sums for $n=7$ .
Answer: 448
Problem: In the adjoining figure, two circles with radii $8$ and $6$ are drawn with their centers $12$ units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$. 

[asy]size(160); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=intersectionpoints(C1,C2)[0]; path C3=Circle(P,sqrt(130)); pair Q=intersectionpoints(C3,C1)[0]; pair R=intersectionpoints(C3,C2)[1]; draw(C1); draw(C2); draw(O2--O1); dot(O1); dot(O2); draw(Q--R); label("$Q$",Q,NW); label("$P$",P,1.5*dir(80)); label("$R$",R,NE); label("12",waypoint(O1--O2,0.4),S);[/asy]
Answer: 130
Problem: The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is $5$, that $BC=6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It follows that the sine of the central angle of minor arc $AB$ is a rational number. If this number is expressed as a fraction $\frac{m}{n}$ in lowest terms, what is the product $mn$?

[asy]size(100); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=1; pair O1=(0,0); pair A=(-0.91,-0.41); pair B=(-0.99,0.13); pair C=(0.688,0.728); pair D=(-0.25,0.97); path C1=Circle(O1,1); draw(C1); label("$A$",A,W); label("$B$",B,W); label("$C$",C,NE); label("$D$",D,N); draw(A--D); draw(B--C); pair F=intersectionpoint(A--D,B--C); add(pathticks(A--F,1,0.5,0,3.5)); add(pathticks(F--D,1,0.5,0,3.5)); [/asy]
Answer: 175
Problem: Find the value of $a_2+a_4+a_6+a_8+\ldots+a_{98}$ if $a_1$, $a_2$, $a_3\ldots$ is an arithmetic progression with common difference 1, and $a_1+a_2+a_3+\ldots+a_{98}=137$.
Answer: 93
Problem: Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is removed, the average of the remaining numbers drops to $55$. What is the largest number that can appear in $S$?
Answer: 649
Problem: Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?
Answer: 24
Problem: The function $f$ is defined on the set of integers and satisfies $f(n)=\begin{cases} n-3&{if}\ n\ge 1000\\ f(f(n+5))&{if}\ n<1000\end{cases}$ Find $f(84)$ .
Answer: 997
Problem: The equation $z^6+z^3+1=0$ has complex roots with argument $\theta$ between $90^\circ$ and $180^\circ$ in the complex plane . Determine the degree measure of $\theta$ .
Answer: 160
Problem: In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is $15\mbox{cm}^2$ and the area of face $ABD$ is $12 \mbox { cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\mbox{cm}^3$.
Answer: 20