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).]

Find all function $ f: R^\plus{} \rightarrow R^\plus{}$ such that for any $ x,y,z \in R^\plus{}$ such that $ x\plus{}y \ge z$ , $ f(x\plus{}y\minus{}z) \plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz}) \equal{} f(x\plus{}y\plus{}z)$

Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2, \ldots, n\}$, let $s(i, j)$ be the number of pairs $(a, b)$ of nonnegative integers satisfying $a i+b j=n$. Let $S$ be the $n$-by-n matrix whose $(i, j)$-entry is $s(i, j)$. For example, when $n=5$, we have $S=\left[\begin{array}{lllll}6 & 3 & 2 & 2 & 2 \\ 3 & 0 & 1 & 0 & 1 \\ 2 & 1 & 0 & 0 & 1 \\ 2 & 0 & 0 & 0 & 1 \\ 2 & 1 & 1 & 1 & 2\end{array}\right]$. Compute the determinant of $S$.

A $k \times k$ array contains each of the numbers $1, 2, \dots, m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k = 3^n$ ($n \in \mathbb{N}^+$)? [i]Attila Sztranyák and Péter Erben[/i]

Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.

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.

Let $t(A)$ denote the sum of elements of a nonempty set $A$ of integers, and define $t(\emptyset)=0$. Find a set $X$ of positive integers such that for every integers $k$ there is a unique ordered pair of disjoint subsets $(A_{k},B_{k})$ of $X$ such that $t(A_{k})-t(B_{k}) = k$.

Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths \[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\] (A triangle is non-degenerate if its vertices are not collinear.) [i]Proposed by Bruno Le Floch, France[/i]

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]

Denote the sequence $a_{i,j}$ in positive reals such that $a_{i,j}$.$a_{j,i}=1$. Let $c_i=\sum_{k=1}^{n}a_{k,i}$. Prove that $1\ge$$\sum_{i=1}^{n}\dfrac {1}{c_i}$

Let $n\ge 2$ be a positive integer. Find the positive integers $x$ \[\sqrt{x+\sqrt{x+\ldots +\sqrt{x}}}<n \] for any number of radicals.

Define the sequences $ a_n, b_n, c_n$ as follows. $ a_0 \equal{} k, b_0 \equal{} 4, c_0 \equal{} 1$. If $ a_n$ is even then $ a_{n \plus{} 1} \equal{} \frac {a_n}{2}$, $ b_{n \plus{} 1} \equal{} 2b_n$, $ c_{n \plus{} 1} \equal{} c_n$. If $ a_n$ is odd, then $ a_{n \plus{} 1} \equal{} a_n \minus{} \frac {b_n}{2} \minus{} c_n$, $ b_{n \plus{} 1} \equal{} b_n$, $ c_{n \plus{} 1} \equal{} b_n \plus{} c_n$. Find the number of positive integers $ k < 1995$ such that some $ a_n \equal{} 0$.

Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?

Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations. [i]Proposed by Warut Suksompong, Thailand[/i]