Determine if for any positive integers $a,b,c$ there exist pairwise distinct non-negative integers $A,B,C$ which are greater than $2019$ such that $a+A,b+B$ and $c+C$ divide $ABC$.

Suppose a function $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x+y)|\geqslant|f(x)+f(y)|$ for all real numbers $x$ and $y$. Prove that equality always holds. Is the conclusion valid if the sign of the inequality is reversed?

On the sides $AB,BC$ and $CA$ of the triangle $ABC$ consider the points $Z,X$ and $Y$ respectively such that \[AZ-AY=BX-BZ=CY-CX.\]Let $P,M$ and $N$ be the circumcenters of the triangles $AYZ, BZX$ and $CXY$ respectively. Prove that the incenters of the triangle $ABC$ coincides with that of the triangle $MNP$.

Determine the greatest positive integer $A{}$ with the following property: however we place the numbers $1,2,\ldots, 100$ on a $10\times 10$ board, each number appearing exactly once, we can find two numbers on the same row or column which differ by at least $A{}$.

An acute triangle $ABC$ in which $AB<AC$ is given. The bisector of $\angle BAC$ crosses $BC$ at $D$. Point $M$ is the midpoint of $BC$. Prove that the line though centers of circles escribed on triangles $ABC$ and $ADM$ is parallel to $AD$.

Let $n$ be a positive integer. Prove that $n^2 + n + 1$ cannot be written as the product of two positive integers of which the difference is smaller than $2\sqrt{n}$.

The sequence $(x_n)_{n\geqslant 0}$ is defined as such: $x_0=1, x_1=2$ and $x_{n+1}=4x_n-x_{n-1}$, for all $n\geqslant 1$. Determine all the terms of the sequence which are perfect squares. George Stoica, Canada

Let $a$ be a positive integer and $(a_n)_{n\geqslant 1}$ be a sequence of positive integers satisfying $a_n<a_{n+1}\leqslant a_n+a$ for all $n\geqslant 1$. Prove that there are infinitely many primes which divide at least one term of the sequence. Moldavia Olympiad, 1994

Let $n$ be a positive integer. In how many ways can we mark cells on an $n\times n$ board such that no two rows and no two columns have the same number of marked cells? Selim Bahadir, Turkey

Let $ABC$ be an acute scalene triangle. The bisector of the angle $\angle ABC$ intersects the altitude $AD$ at $K$. Let $M$ be the projection of $B$ onto $CK$ and let $N$ be the intersection between $BM$ and $AK$. Let $T$ be a point on $AC$ such that $NT$ is parallel to $DM$. Prove that $BM$ is the bisector of the angle $\angle TBC$. Melih Üçer, Turkey