sungyub/dapo-math-17k-verl huggingface dataset:
user: For real numbers $B$, $M$, and $T$, we have the following conditions: 
$B^2 + M^2 + T^2 = 2022$ 
$B + M + T = 72$

Compute the sum of the minimum and maximum possible values of $T$.

Answer: 48
user: Kara rolls a six-sided die six times, and notices that the results satisfy the following conditions:
- She rolled a $6$ exactly three times.
- The product of her first three rolls is the same as the product of her last three rolls.

How many distinct sequences of six rolls could Kara have rolled?

Answer: 72
user: How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?

Answer: 28
user: The product $55 \cdot 60 \cdot 65$ is written as a product of 5 distinct numbers. Find the least possible value of the largest number among these 5 numbers.

Answer: 20
user: $N$ is a 50-digit number in decimal representation. All digits except the 26th digit (from the left) are 1. If $N$ is divisible by 13, find its 26th digit.

Answer: 3
user: A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer?

Answer: 13
user: For each positive integer $n$, consider the highest common factor $h_n$ of the two numbers $n! + 1$ and $(n+1)!$. For $n < 100$, find the largest value of $h_n$.

Answer: 97
user: Each of the 2009 distinct points in the plane is colored either blue or red. On every unit circle centered at a blue point, there are exactly two red points. Find the greatest possible number of blue points.

Answer: 1964
user: Determine the largest positive integer $n$ for which there exist pairwise different sets $\mathbb{S}_1 , \ldots, \mathbb{S}_n$ with the following properties:

$|\mathbb{S}_i \cup \mathbb{S}_j| \leq 2004$ for any two indices $1 \leq i, j \leq n$.
$\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1, 2, \ldots, 2008 \}$ for any $1 \leq i < j < k \leq n$.

Answer: 32
user: Princess Pear has $100$ jesters with heights $1, 2, \dots, 100$ inches. On day $n$ with $1 \leq n \leq 100$, Princess Pear holds a court with all her jesters with height at most $n$ inches, and she receives two candied cherries from every group of $6$ jesters with a median height of $n - 50$ inches. A jester can be part of multiple groups.

On day $101$, Princess Pear summons all $100$ jesters to court one final time. Every group of $6$ jesters with a median height of $50.5$ inches presents one more candied cherry to the Princess. How many candied cherries does Princess Pear receive in total?

Please provide a numerical answer (with justification).

Answer: 384160000
user: Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.

Answer: 2077
user: From a list of integers from $1$ to $2022$, inclusive, delete all numbers in which at least one of its digits is a prime. How many numbers remain without erasing?

Answer: 431
user: We say that a positive integer $m$ covers the number $1998$ if the digits $1, 9, 9, 8$ appear in this order as digits of $m$. For instance, $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.

Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n \ge 5$), with all digits different from $0$. What is the remainder of $k(n)$ when divided by $8$?

Answer: 1
user: The sum
\[
\frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!}
\]
can be expressed as a rational number \(N\). Find the last 3 digits of \(2021! \cdot N\).

Answer: 977
user: How many positive integers less than 2005 are relatively prime to 1001?

Answer: 1442
user: There are $2017$ points on the plane, no three of them are collinear. Some pairs of the points are connected by $n$ segments. Find the smallest value of $n$ so that there always exists two disjoint segments in any case.

Answer: 2017
user: Let $f(n) = 1 \times 3 \times 5 \times \cdots \times (2n-1)$. Compute the remainder when $f(1) + f(2) + f(3) + \cdots + f(2016)$ is divided by $100$.

Answer: 24
user: There are four entrances into Hades. Hermes brings you through one of them and drops you off at the shore of the river Acheron where you wait in a group with five other souls, each of whom had already come through one of the entrances as well, to get a ride across. In how many ways could the other five souls have come through the entrances such that exactly two of them came through the same entrance as you did? The order in which the souls came through the entrances does not matter, and the entrance you went through is fixed.

Answer: 270
user: Find the coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) \cdots (1 - 2008x)$.

Answer: -1004
user: The integer $n$, between 10000 and 99999, is represented as $abcde$ in decimal notation. The digits $a$, $b$, $c$, $d$, and $e$ are defined as follows:

- $a$ is the remainder when $n$ is divided by 2.
- $b$ is the remainder when $n$ is divided by 3.
- $c$ is the remainder when $n$ is divided by 4.
- $d$ is the remainder when $n$ is divided by 5.
- $e$ is the remainder when $n$ is divided by 6.

Find the value of $n$.

Answer: 11311
user: Let $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ is divided by $1000$.

Answer: 167
user: In the $3$-dimensional coordinate space, find the distance from the point $(36, 36, 36)$ to the plane that passes through the points $(336, 36, 36)$, $(36, 636, 36)$, and $(36, 36, 336)$.

Answer: 200
user: How many solutions does the equation \( \frac{\lfloor x \rfloor}{\{ x \}} = \frac{2007x}{2008} \) have?

Answer: 2006
user: $\text{HOW}$, $\text{BOW}$, and $\text{DAH}$ are equilateral triangles in a plane such that $WO = 7$ and $AH = 2$. Given that $D$, $A$, $B$ are collinear in that order, find the length of $BA$.

Answer: 11
user: In how many ways can each of the integers $1$ through $11$ be assigned one of the letters $L$, $M$, and $T$ such that consecutive multiples of $n$, for any positive integer $n$, are not assigned the same letter?

Answer: 24
user: Find the least positive integer $N$ such that the only values of $n$ for which $1 + N \cdot 2^n$ is prime are multiples of $12$.

Answer: 556
user: A $1 \times 5$ rectangle is divided into five unit squares (cells) numbered 1 through 5 from left to right. A frog starts at cell 1. Every second, it jumps from its current cell to one of the adjacent cells. The frog makes exactly 14 jumps. How many paths can the frog take to finish at cell 5?

Answer: 364
user: Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0 \leq n < 16} 2^n (-1)^{s(n)}. \]

Answer: 45
user: 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
user: 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
user: 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
user: 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
user: Find all real numbers $x$ such that $4x^5 - 7$ and $4x^{13} - 7$ are both perfect squares.

Answer: 2
user: 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
user: 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
user: 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
user: 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
user: 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
user: Find all prime numbers $p$ such that $5^p + 4p^4$ is a perfect square.

Answer: 5
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: Find the sum of all integers $n$ satisfying the following inequality:

\[
\frac{1}{4}<\sin\frac{\pi}{n}<\frac{1}{3}.
\]

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

$f(x)$ 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
user: 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
user: 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
user: 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
user: 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:

Each pair of students is in exactly one club.
For each student and each society, the student is in exactly one club of the society.
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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: Find all integers $n$ such that $n^{n-1} - 1$ is square-free.

Answer: 2
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: Determine all integers $n > 1$ whose positive divisors add up to a power of $3$.

Answer: 2
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: Let $a$ and $b$ be positive integers such that $2a - 9b + 18ab = 2018$. Find $b - a$.

Answer: 223
user: 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
user: Determine the minimum possible amount of distinct prime divisors of $19^{4n} + 4$, where $n$ is a positive integer.

Answer: 3
user: 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
user: 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