An [i]anti-Pascal[/i] triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from $1$ to $10$. \[\begin{array}{ c@{\hspace{4pt}}c@{\hspace{4pt}} c@{\hspace{4pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c } \vspace{4pt} & & & 4 & & & \\\vspace{4pt} & & 2 & & 6 & & \\\vspace{4pt} & 5 & & 7 & & 1 & \\\vspace{4pt} 8 & & 3 & & 10 & & 9 \\\vspace{4pt} \end{array}\] Does there exist an anti-Pascal triangle with $2018$ rows which contains every integer from $1$ to $1 + 2 + 3 + \dots + 2018$? [i]Proposed by Morteza Saghafian, Iran[/i]

A sequence of numbers $a_n, n = 1,2, \ldots,$ is defined as follows: $a_1 = \frac{1}{2}$ and for each $n \geq 2$ \[ a_n = \frac{2 n - 3}{2 n} a_{n-1}. \] Prove that $\sum^n_{k=1} a_k < 1$ for all $n \geq 1.$

Let $S$ be a set of nonnegative integers such that [list] [*] there exist two elements $a$ and $b$ in $S$ such that $a,b>1$ and $\gcd(a,b)=1$; and [*] for any (not necessarily distinct) element $x$ and nonzero element $y$ in $S$, both $xy$ and the remainder when $x$ is divided by $y$ are in $S$. [/list] Prove that $S$ contains every nonnegative integer. [i]Jacob Paltrowitz[/i]

We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black. Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black? [It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]

In each square of a garden shaped like a $2022 \times 2022$ 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: [list] [*] 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. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

Denote by $\mathbb{Q}^+$ the set of positive rational numbers. A function $f : \mathbb{Q}^+ \to \mathbb{Q}$ satisfies • $f(p) = 1$ for all primes $p$, and • $f(ab) = af(b) + bf(a)$ for all $ a,b \in \mathbb{Q}^+ $. For which positive integers $n$ does the equation $nf(c) = c$ have at least one solution $c$ in $\mathbb{Q}^+$?

Prove that if $ n $ is an integer greater than $ 4 $, then $ 2^n $ is greater than $ n^2 $.

Let $\mathcal{F}$ be the set of all functions $f : (0,\infty)\to (0,\infty)$ such that $f(3x) \geq f( f(2x) )+x$ for all $x$. Find the largest $A$ such that $f(x) \geq A x$ for all $f\in\mathcal{F}$ and all $x$.

Let $ n$ be a positive integer. Show that \[ \left(\sqrt{2} \plus{} 1 \right)^n \equal{} \sqrt{m} \plus{} \sqrt{m\minus{}1}\] for some positive integer $ m.$

Let $f(x)=\frac{1+\cos(2 \pi x)}{2}$, for $x \in \mathbb{R}$, and $f^n=\underbrace{ f \circ \cdots \circ f}_{n}$. Is it true that for Lebesgue almost every $x$, $\lim_{n \to \infty} f^n(x)=1$?

An $ (n, k) \minus{}$ tournament is a contest with $ n$ players held in $ k$ rounds such that: $ (i)$ Each player plays in each round, and every two players meet at most once. $ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$. Determine all pairs $ (n, k)$ for which there exists an $ (n, k) \minus{}$ tournament. [i]Proposed by Carlos di Fiore, Argentina[/i]

Assume we are given a set of weights, $x_1$ of which have mass $d_1$, $x_2$ have mass $d_2$, etc, $x_k$ have mass $d_k$, where $x_i,d_i$ are positive integers and $1\le d_1<d_2<\ldots<d_k$. Let us denote their total sum by $n=x_1d_1+\ldots+x_kd_k$. We call such a set of weights [i]perfect[/i] if each mass $0,1,\ldots,n$ can be uniquely obtained using these weights. (a) Write down all sets of weights of total mass $5$. Which of them are perfect? (b) Show that a perfect set of weights satisfies $$(1+x_1)(1+x_2)\cdots(1+x_k)=n+1.$$ (c) Conversely, if $(1+x_1)(1+x_2)\cdots(1+x_k)=n+1$, prove that one can uniquely choose the corresponding masses $d_1,d_2,\ldots,d_k$ with $1\le d_1<\ldots<d_k$ in order for the obtained set of weights is perfect. (d) Determine all perfect sets of weights of total mass $1993$.

If $x_0=x_1=1$, and for $n\geq1$ $x_{n+1}=\frac{x_n^2}{x_{n-1}+2x_n}$, find a formula for $x_n$ as a function of $n$.

Let $A = \{a_1, \dots, a_{2024}\}$ be a set of $2024$ pairwise distinct real numbers. Assume that there exist positive integers $b_1, b_2,\dotsc,b_{2024}$ such that \[ a_1b_1 + a_2b_2 + \dots + a_{2024}b_{2024} = 0. \] Prove that one can choose $a_{2025}, a_{2026}, a_{2027}, \dots$ such that $a_k \in A$ for all $k \ge 2025$ and, for every positive integer $d$, there exist infinitely many positive integers $n$ satisfying \[ \sum_{k=1}^n a_k k^d = 0. \] [i]Daniel Zhu[/i]

$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$