Let $\mathbb N$ be the set of positive integers. Find all functions $f\colon\mathbb N\to \mathbb N$ such that for every $m,n\in \mathbb N$, \[ f(m)+f(n)\mid m+n. \]

Let $ABC$ be an acute triangle. Let $M_A, M_B$ and $M_C$ be the midpoints of sides $BC,CA$, respectively $AB$. Let $M'_A , M'_B$ and $M'_C$ be the the midpoints of the arcs $BC, CA$ and $AB$ respectively of the circumscriberd circle of triangle $ABC$. Let $P_A$ be the intersection of the straight line $M_BM_C$ and the perpendicular to $M'_BM'_C$ through $A$. Define $P_B$ and $P_C$ similarly. Show that the straight line $M_AP_A, M_BP_B$ and $M_CP_C$ intersect at one point.

We are given $n$ distinct rectangles in the plane. Prove that between the $4n$ interior angles formed by these rectangles at least $4\sqrt n$ are distinct.

Let $\varphi$ denote the Euler phi-function. Prove that for every positive integer $n$ $$2^{n(n+1)} | 32 \cdot \varphi \left( 2^{2^n} - 1 \right).$$

Find all the positive integers $a, b, c$ such that $$a! \cdot b! = a! + b! + c!$$

Let $n \ge 2$ be an integer. Consider the following game: Initially, $k$ stones are distributed among the $n^2$ squares of an $n\times n$ chessboard. A move consists of choosing a square containing at least as many stones as the number of its adjacent squares (two squares are adjacent if they share a common edge) and moving one stone from this square to each of its adjacent squares. Determine all positive integers $k$ such that: (a) There is an initial configuration with $k$ stones such that no move is possible. (b) There is an initial configuration with $k$ stones such that an infinite sequence of moves is possible.

Let $ABCD$ be an isosceles trapezoid with bases $AD> BC$. Let $X$ be the intersection of the bisectors of $\angle BAC$ and $BC$. Let $E$ be the intersection of$ DB$ with the parallel to the bisector of $\angle CBD$ through $X$ and let $F$ be the intersection of $DC$ with the parallel to the bisector of $\angle DCB$ through $X$. Show that quadrilateral $AEFD$ is cyclic.

Let $n$ be a positive integer . Let $x_1 \le x_2 \le \dots \le x_{m}$ be a sequence of real numbers such that $x_{1}+x_{2}+\cdots+x_{n}=0 $ and $ x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1$. Prove that $x_1x_n \leqslant-\frac{1}{n}$. Click to reveal hidden textSwitzerland 2020 Swiss MO p8 wording