sungyub/dapo-math-17k-verl huggingface dataset:
user: 在数列 $\left\{a_{n}\right\}$ 中, $a_{1}=1$, 当 $n \geqslant 2$ 时, $a_{n} 、 S_{n} 、 S_{n}-\frac{1}{2}$ 成等比数列。则 $\lim _{n \rightarrow \infty} n^{2} a_{n}=$ \_\_\_\_\_。原始答案是以\frac{m}{n}的形式，请给出m + n的值。

Answer: 1
user: 已知在三棱锥 $D-A B C$ 中, $\angle A C B=\angle A B D=90^{\circ}, C A=C B, \angle B A D=30^{\circ}$.若点 $C$ 在平面 $A B D$ 上的射影恰好在 $A D$ 上, 则二面角 $C-A B-D$ 的平面角的正弦值大小为 \frac{\sqrt{m}}{n}，请给出m + n的值。

Answer: 9
user: 设函数 $f(x)=x-\ln (a x+2 a+1)+2$. 若对于任意 $x \geqslant-2$, 均有 $f(x) \geqslant 0$ 恒成立，请给出实数 $a$ 的取值范围端点之和。

Answer: 1
user: 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 $x_{2}$为一个分数$\frac{m}{n}$的形式，请给出m + n的值。

Answer: 3
user: 设函数
$$f(x)=x-\ln(ax+2a+1)+2.$$ 
若对于任意的 $x\geqslant-2$,均有 $f(x)\geqslant0$,则实数$a$的取值范围是\qquad. 请提供区间端点之和。

Answer: 1
user: 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，给出其方程中 x 的系数值。

Answer: 1
user: 在三棱锥 $D-A B C$ 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
user: 已知实数 $x, y$ 满足 $x|x|+\frac{y|y|}{3}=1$, 则 $|\sqrt{3} x+y-4|$ 的取值范围为一个区间$[p, q)$，请给出p + q的整数部分的值。

Answer: 8
user: 若 $\triangle A B C$ 的三边 $a, b, c$ 满足 $a^{2}+b^{2}+3 c^{2}=7$, 则 $\triangle A B C$ 面积的最大值为。原始答案为\frac{\sqrt{m}}{n}的格式，请给出m + n的值。

Answer: 11
user: 若 $\triangle A B C$ 的三边 $a 、 b 、 c$ 满足 $a^{2}+b^{2}+3 c^{2}=7$ ，则 $\triangle A B C$ 的面积的最大值是。原始答案为\frac{\sqrt{m}}{n}，请给出m + n的值。

Answer: 11
user: 给定正整数 $a, b(a \leqslant b)$. 数列 $\left\{f_{n}\right\}$ 满足: $$ f_{1}=a, f_{2}=b, f_{n+2}=f_{n+1}+f_{n}(n=1,2, \cdots) $$ 若对任意的正整数 $n$, 都有 $\left(\sum_{k=1}^{n} f_{k}\right)^{2} \leqslant \lambda \cdot f_{n} f_{n+1}$, 求实数 $\lambda$ 的最小值. 原始答案为 $m + \sqrt{n}$ 的形式，请给出 m + n 的值.

Answer: 7
user: 设 $A, B, C$ 是 $\triangle A B C$ 的三个内角, 则 $3 \cos A+2 \cos 2 B+\cos 3 C$ 的取值范围为. 请给出该范围的端点之和的整数部分。

Answer: 4
user: 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: $$ \sum_{k=1}^{n} a_{2 k-1} \cdot \sum_{k=1}^{n} a_{2 k}=\prod_{k=1}^{n} a_{2 k-1}+\prod_{k=1}^{n} a_{2 k}. $$ 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 $n = 2$ 时的最小值。

Answer: 8
user: 已知 $f(x)$ 是定义在 $\mathbf{R}$ 上的函数, 且对任意实数 $x$, 均有 $2 f(x)+f\left(x^{2}-1\right)=1$, 请给出 $f(\sqrt{2})$ 的值为\(\frac{m}{n}\)的格式，并计算m + n的值。

Answer: 4
user: 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 $\triangle P Q R$ 的面积的最小值是以 \( \frac{\sqrt{m} \cdot e^n}{k} \) 的形式表示，请给出 $m + n + k$ 的值。

Answer: 5
user: 给定正整数 $a, b(a \leqslant b)$. 数列 $\left\{f_{n}\right\}$ 满足: $$ f_{1}=a, f_{2}=b, f_{n+2}=f_{n+1}+f_{n}(n=1,2, \cdots) $$ 若对任意的正整数 $n$, 都有 $\left(\sum_{k=1}^{n} f_{k}\right)^{2} \leqslant \lambda \cdot f_{n} f_{n+1}$, 求实数 $\lambda$ 的最小值。原始答案为 $n+\sqrt{m}$，请给出 $n+m$ 的值。

Answer: 7
user: 已知锐角 \( \angle A 、 \angle B \) 满足 \( \tan (A+B)=2 \tan A \). \n则 \( \tan B \) 的最大值是一个分数 \( \frac{m}{n} \) 的形式，请给出 \( m + n \) 的值。

Answer: 6
user: 如图1，已知正方形 ABCD 的边长为 3 $E$为$DC$ 的中点$,AE$ 与 $BD$交于点 $F.$ 则$
\overrightarrow{FD}\cdot\overrightarrow{DE}$的原始答案为$-\frac{m}{n}$格式，请给出m + n的值。

Answer: 5
user: 设等比数列 ${an}$的前$n$项和 为$S_n$ ，且$a_{n+1}=2S_{n}+\frac{1}{2}(n\in\mathbf{Z}_{+}).$\求数列 ${an}$的通项公式。已知原始答案为$\frac{a}{b}\times c^{n-1}$的形式，请给出$a + b + c$的值。

Answer: 6
user: 用 6 根等长的细铁棒焊接成一个正四面体形框架, 铁棒的粗细和焊接误差不计。设此框架能容纳得下的最大球的半径为 $R_1$,能包容此框架的最小球的半径为 $R_2$. 则原始答案为\frac{\sqrt{m}}{n}的格式，请给出m + n的值。

Answer: 6
user: 已知直线 $l: y=\sqrt{3} x+4$, 动圆 $O: x^{2}+y^{2}=r^{2}(1<r<2)$, 菱形 $A B C D$ 的一个内角为 $60^{\circ}$, 顶点 $A, B$ 在直线 $l$ 上, 顶点 $C, D$ 在圆 $O$ 上, 当 $r$ 变化时, 求菱形 $A B C D$ 的面积 $S$ 的取值范围. 计算该取值范围的端点之和，并将结果四舍五入为最接近的整数.

Answer: 15
user: 若 $a 、 b 、 c$ 成等差数列, 则直线 $a x+b y+c=0$ 被椭圆 $\frac{x^{2}}{2}+\frac{y^{2}}{8}=1$ 截得线段的中点的轨迹方程为某椭圆方程。请给出该椭圆方程中x与y的系数和。

Answer: -1
user: 如图，圆$C$与$x$轴相切于点$T(2,0)$，与$y$轴的正半轴相交于$A, B$两点（$A$在$B$的上方），且$|AB|=3$。求圆$C$的方程。已知圆的方程为$(x-m)^{2}+(y-\frac{n}{k})^{2}=\frac{p}{q}$，请给出m + n + k的值。

Answer: 9
user: 设 $O$ 为 $\triangle A B C$ 的外心, 且满足$$\begin{aligned}& \overrightarrow{O A}+\overrightarrow{O B}=\overrightarrow{O C}. \\
& \text { 则 } \angle A C B=\end{aligned}$$

原始答案为 $\frac{m\pi}{n}$ 格式，请给出 m + n 的值。

Answer: 5
user: 某人抛郑一枚硬币，出现正面向上和反面向上的概率均为 $\frac{1}{2}$. 构造数列 $\left\{a_{n}\right\}$ ，使\na_{n}= \begin{cases}1, & \text { 第 } n \text { 次正面向上; } \\ -1, & \text { 第 } n \text { 次反面向上. }\end{cases}\n记 $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$. 则 $S_{2} \neq 0$, 且 $S_{8}=2$的概率为\n原始答案为 \(\frac{m}{n}\) 的格式，请给出 m + n 的值。

Answer: 141
user: A $2\times3$ rectangle has vertices at $(0,0)$, $(2,0)$, $(0,3)$, and $(2,3)$. It rotates $90^{\circ}$ clockwise about the point $(2,0)$. It then rotates $90^{\circ}$ clockwise about the point $(5,0)$, then $90^{\circ}$ clockwise about the point $(7,0)$, and finally, $90^{\circ}$ clockwise about the point $(10,0)$. (The side originally on the x-axis is now back on the x-axis.) Find the area of the region above the x-axis and below the curve traced out by the point whose initial position is $(1,1)$. The original answer is in the format $\frac{m\pi}{n} + k$. Please give the value of m + n + k.

Answer: 15
user: Let P be a polynomial, with real coefficients, in three variables and F be a function of two variables such that

$P(ux,uy, uz)=u^{2}F(y-x,z-x,z-x)$ for all real $x$, $y$, $z$, $u$,

and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let A, B, C be complex numbers with $P(A,B,C)=0$ and $|B- A|=10$. Find $|C-A|$. The original answer is in the format $(m/k)\sqrt{n}$, please give the value of m + n + k.

Answer: 38
user: Find the maximum value of \(\int_0^y \sqrt{x^4+(y-y^2)^2} \, dx\) for \(0 \leq y \leq 1\). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).

Answer: 4
user: For each positive integer $n$, let $a(n)$ be the number of zeros in the base 3 representation of $n$. For which positive real numbers $x$ does the series $$ \sum_{n=1}^{\infty} \frac{x^{a(n)}}{n^3} $$ converge? Please provide the sum of the endpoints of the interval for which the series converges.

Answer: 25
user: Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points? (It is understood that each point is independently chosen relative to a uniform distribution on the sphere.) The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 9
user: Let $S$ be the smallest set of positive integers such that\n(a) 2 is in $S$,\n(b) $n$ is in $S$ whenever $n^2$ is in $S$, and\n(c) $(n+5)^2$ is in $S$ whenever $n$ is in $S$.\nWhich positive integers are not in $S$? The original answer is in the form of a set, please provide the sum of all positive integers not in $S$.\n

Answer: 6
user: Find all ordered pairs $(a, b)$ of positive integers for which
$$
\frac{1}{a}+\frac{1}{b}=\frac{3}{2018} .
$$
Please provide the sum of all integers in the ordered pairs.

Answer: 1438383
user: Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that $$\begin{pmatrix}m\\n-1\end{pmatrix}>\begin{pmatrix}m-1\\n\end{pmatrix}.$$ Evaluate the limit $$\lim_{n\to\infty}\frac{M(n)}{n}.$$ The original answer is in the format $\frac{a + \sqrt{b}}{c}$. Please provide the integer sum of $a$, $b$, and $c$.

Answer: 10
user: Let $k$ be a nonnegative integer. Evaluate $$\sum_{j=0}^k2^{k-j}\binom{k+j}j.$$ Please provide the value of the expression when $k = 1$. The original answer is in $4^{k}$ format, please give the value of $m + n$

Answer: 5
user: For every positive real number $x$, let
$$g(x)=\lim\limits_{r\to0}((x+1)^{r+1}-x^{r+1})^{\frac{1}{r}}.$$\mathrm{Find}\lim_{x\to\infty}\frac{g(x)}{x}. The original answer is in the form of $e$, please provide the integer part of the result.

Answer: 1
user: Let $n$ be an integer with $n\geq2.$ Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of negative coefficients of $p(x)^2?$ Provide the value when $n=2$. directly give the value of the answer when n=4

Answer: 6
user: Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n;$ that is to say, $p(x)=x^{2n}+a_{2n-1}x^{2n-1}+\cdots+a_{1}x+a_{0}$ for some real coefficients $a_{0},\ldots,a_{2n-1}.$ Suppose that $p(1/k)=k^{2}$ for all integers $k$ such that $1\leq|k|\leq n.$ Find all other real numbers $x$ for which $p(1/x)=x^2.$ The original answer is in the form of $\pm m/n!$, please give the value of m+ n

Answer: 2
user: Let $a_0=5/2$ and $a_k=a^2_{k-1}-2$ for $k\geq1.$ Compute $$\prod\limits_{k=0}^{\infty}\left(1-\frac{1}{a_k}\right)$$ in closed form. The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 10
user: Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that $$f(x)+f\left(1-\dfrac{1}{x}\right)=\arctan x$$ for all real $x\neq0.$ (As usual, y=arctan $x$ means $-\pi/2<$ $y<\pi/2$ and tan$y=x.)$ Find $$\int_0^1f(x)\:dx.$$ The original answer is in the format \(\frac{m \pi}{n}\). Please give the value of \(m + n\).

Answer: 11
user: Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k\leqslant n.$ The rules of the game are: $(i)$ $A$ player cannot choose a number that has been chosen by either player on any previous turn. $(ii)$ $A$ player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. $(iii)$ The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the sum of all $n$ for which the game ends in a draw, assuming that both players use optimal strategies.

Answer: 13
user: Determine all triples $(a,b,c)$ of positive integers for which $ab-c,bc-a$, and $ca-b$ are powers of $2$. Explanation:$A$ power of $2$ is an integer of the form $2^n$, where $n$ denotes some nonnegative integer. 
How many triples $(a, b, c)$ satisfy the conditions?

Answer: 16
user: Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ is called $k$-good if $\gcd(f(m)+n,f(n)+m)\leqslant k$ for all $m\neq n.$ Find the smallest positive integer $k$ such that there exists a $k$-good function.

Answer: 2
user: Determine the largest real number $a$ such that for all $n\geqslant1$ and for all real numbers $x_0,x_1,\ldots,x_n$ satisfying $0=x_0<x_1<x_2<\cdots<x_n$, we have $$\begin{aligned}\frac{1}{x_1-x_0}+\frac{1}{x_2-x_1}+\cdots+\frac{1}{x_n-x_{n-1}}\geqslant a\left(\frac{2}{x_1}+\frac{3}{x_2}+\cdots+\frac{n+1}{x_n}\right).\end{aligned}\quad(1)$$ The largest $a$ is in the format of $\frac{m}{n}$. Please provide the value of$ m + n$.

Answer: 13
user: The leader of an IMO team chooses positive integers $n$ and $k$ with $n>k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader's in exactly $k$ positions. (For example, if $n=3$ and $k=1$, and if the leader chooses $101$, the deputy leader would write down $001,111$and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader's string. What is the minimum number of guesses (in terms of $n$ and $k)$ needed to guarantee the correct answer? Please give the sum of guesses when $n = 6$ and $k = 3$.

Answer: 2
user: Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n).$ Please provide the value of $f(n)$ when $n=2$. Directly give the value of $f(n)$.

Answer: 4
user: Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: $\bullet$ In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin. $\bullet$ In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the $frst\textit{line. }$ $\bullet$ In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a,b,c,d$ are four (not necessarily distinct) numbers from the $frst\textit{line. }$ Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line. Please provide the sum of all possible values of $q$.

Answer: 0
user: For each permutation $y_1,y_2,\cdots,y_n$ of these numbers we have $\sum\limits_{i=1}^{n-1}y_iy_{i+1}=y_1y_2+y_2y_3+y_3y_4+\cdots+y_{n-1}y_n\geqslant-1$. Find the largest constant $K=K(n)$ such that $\sum\limits_{1\leqslant i<j\leqslant n}x_ix_j\geqslant K$ holds for every Shiny$n$-tuple $(x_1,x_2,\cdots,x_n)$. Please provide the value of $K$ when $n=3$.

Answer: 1
user: Sir Alex plays the following game on a row of $9$ cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: (1) Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. (2) Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j.$ Replace the number in one of the cells with $2^j+1$ and erase the number in the other cell. At the end of the game, one cell contains the number $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. Please provide the value of the maximum number of moves when $n = 10$.

Answer: 2025
user: Let $n>1$ be an integer. An $n\times n\times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one color. For each $n\times n\times1$ box consisting of $n^2$ unit cubes (of any of the three possible orientations), we consider the set of the colors present in that box (each color is listed only once). This way, we get $3n$ sets of colors, split into three groups according to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colors that are present. Please provide the maximum number of colors when n = 6.

Answer: 91
user: Let $n$ be a given positive integer. In the Cartesian plane, each lattice point with nonnegative coordinates initially contains a butterfly, and there are no other butterflies. The neighborhood of a lattice point $c$ consists of all lattice points within the axis-aligned $(2n+1)\times (2n+1)$ square centered at $c$, apart from $c$ itself. We call a butterfly $lonely, crowded$, or $comfortable$, depending on whether the number of butterflies in its neighborhood $N$ is respectively less than, greater than, or equal to half of the number of lattice points in $N.$ Every minute, all lonely butterflies fly away simultaneously. This process goes on for as long as there are any lonely butterflies. Assuming that the process eventually stops, determine the number of comfortable butterflies at the final state. Please provide the number of comfortable butterflies when $n = 3$.

Answer: 10
user: Find all pairs $(p,q)$ of prime numbers with $p>q$ for which the number $$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$ is an integer. Please provide the value of $p+q$ for such pairs.

Answer: 5
user: Let $a_0, a_1, a_2, \ldots$ be a sequence of real numbers such that $a_0=0, a_1=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying
$$
a_n=\frac{a_{n-1}+\cdots+a_{n-k}}{k}
$$
Find the maximal possible value of $a_{2018}-a_{2017}$. The original answer is in the form \frac{m}{n^2}, please give the value of $m + n$.

Answer: 4033
user: Let $k$ be a positive integer. The organising committee of a tennis tournament is to schedule the matches for $2 k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay 1 coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.Write down the value when $k=4$.

Answer: 134
user: A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold: (i) each triangle from $T$ is inscribed in $\omega$; (ii) no two triangles from $T$ have a common interior point. Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$. Please provide the maximum value of $t$.

Answer: 4
user: Find all pairs $(m, n)$ of positive integers satisfying the equation $$ \begin{equation*} \left(2^{n}-1\right)\left(2^{n}-2\right)\left(2^{n}-4\right) \cdots\left(2^{n}-2^{n-1}\right)=m! \tag{1} \end{equation*} $$
Please provide the sum of all integers in the pairs $(m, n)$.

Answer: 7
user: Find all triples $(a, b, c)$ of positive integers such that $a^{3}+b^{3}+c^{3}=(a b c)^{2}$. Please provide the sum of all integers in the solutions.

Answer: 18
user: Version 1. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, $$\sqrt[N]{\frac{x^{2N}+1}{2}}\leqslant a_n(x-1)^2+x.$$ Please provide the value of N when $a_n = N/2$.\n\nVersion 2. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, $$\sqrt[N]{\frac{x^{2N}+1}{2}}\leqslant b_N(x-1)^2+x.$$ Please provide the sum of $a_{4}$$b_{6}$.

Answer: 5
user: Let $n$ be an integer with $n \geqslant 2$. On a slope of a mountain, $n^2$ checkpoints are marked, numbered from 1 to $n^2$ from the bottom to the top. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars numbered from 1 to $k$; each cable car provides a transfer from some checkpoint to a higher one. For each company, and for any $i$ and $j$ with $1 \leqslant i<j \leqslant k$, the starting point of car $j$ is higher than the starting point of car $i$; similarly, the finishing point of car $j$ is higher than the finishing point of car $i$. Say that two checkpoints are linked by some company if one can start from the lower checkpoint and reach the higher one by using one or more cars of that company (no movement on foot is allowed). Determine the smallest $k$ for which one can guarantee that there are two checkpoints that are linked by each of the two companies. Please provide the value of $k$ when $n=3$.

Answer: 7
user: Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $\left(d_1, d_2, \ldots, d_k\right)$ such that for every $i=1,2, \ldots, k$, the number $d_1+\cdots+d_i$ is a perfect square. Please provide the sum of all such $n$.

Answer: 4
user: Find all positive integers $n\geqslant2$ for which there exist $n$ real numbers $a_{1}<\cdots<a_{n}$ and a real number $r>0$ such that the $\frac{1}{2}n(n-1)$ differences $a_{j}-a_{i}$ for $1\leqslant i<j\leqslant n$ are equal, in some order, to the numbers $r^{1},r^{2},\ldots,r^{\frac{1}{2}n(n-1)}$. Please provide the sum of all such $n$ values.

Answer: 9
user: In each square of a garden shaped like a $2022\times2022$ board, there is initially a tree of height 0.A gardener and a lumberjack alternate turns playing the following game,with the gardener taking the first turn:

• The gardener chooses a square in the garden.Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. 
• The lumberjack then chooses four different squares on the board.Each tree of positive height on those squares then becomes one unit shorter.

We say that a tree is majestic if its height is at least $10^6$ .Determine the largest number $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board,no matter how the lumberjack plays.

Answer: 2271380
user: Let $n>3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbours on the right and left. Determine all initial distributions of coins from which it is possible that, after a finite number of steps, each child has exactly one coin. Please provide the value when $n=5$.

Answer: 15
user: Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. For positive integer $n=8$, ﬁnd the smallest number of non-empty piles that one can obtain by performing a ﬁnite sequence of moves of this form.

Answer: 1
user: Let $\mathbb{Z}_{\geqslant 0}$ be the set of non-negative integers, and let $f: \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ be a bijection such that whenever $f\left(x_1, y_1\right)>f\left(x_2, y_2\right)$, we have $f\left(x_1+1, y_1\right)>f\left(x_2+1, y_2\right)$ and $f\left(x_1, y_1+1\right)>f\left(x_2, y_2+1\right)$.\n\nLet $N$ be the number of pairs of integers $(x, y)$, with $0 \leqslant x, y<100$, such that $f(x, y)$ is odd. Find the smallest and largest possible value of $N$. Please provide the sum of these two values.

Answer: 10000
user: Find all triples of positive integers $(a, b, p)$ with $p$ prime such that $a^p=b!+p$. Please provide the sum of all elements in these triples.

Answer: 16
user: 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 value of $n+1$ if the answer can be expressed in the form $2^{n} - 1$.

Answer: 2025
user: 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 value of $k$ when $n = 6$.

Answer: 3
user: Let $n \geqslant 2$ be a positive integer. Paul has a $1 \times n^{2}$ rectangular strip consisting of $n^{2}$ unit squares, where the $i^{\text {th }}$ square is labelled with $i$ for all $1 \leqslant i \leqslant n^{2}$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then translate (without rotating or flipping) the pieces to obtain an $n \times n$ square satisfying the following property: if the unit square in the $i^{\text {th }}$ row and $j^{\text {th }}$ column is labelled with $a_{i j}$, then $a_{i j}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this. Please provide the value of the smallest number of pieces when $n=3$.

Answer: 5