In a group of kids there are $2022$ boys and $2023$ girls. Every girl is a friend with exactly $2021$ boys. Friendship is a symmetric relation: if A is a friend of B, then B is also a friend of A. Prove that it is not possible that all boys have the same number of girl friends. Proposed by the JMMO Problem Selection Committee

A positive integer is called superprime if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime integers. Proposed by Nikola Velov

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality $$ \left ( \frac{1+a}{b}+2 \right ) \left ( \frac{1+b}{c}+2 \right ) \left ( \frac{1+c}{a}+2 \right )\geq 216.$$When does equality hold? Proposed by Anastasija Trajanova

We are given an acute $\triangle ABC$ with circumcenter $O$ such that $BC<AB$. The bisector of $\angle ACB$ meets the circumcircle of $\triangle ABC$ at a second point $D$. The perpendicular bisector of $AC$ meets the circumcircle of $\triangle BOD$ for the second time at $E$. The line $DE$ meets the circumcircle of $\triangle ABC$ for the second time at $F$. Prove that the lines $CF$, $OE$ and $AB$ are concurrent. Proposed by Petar Filipovski

Consider a $2023\times2023$ board split into unit squares. Two unit squares are called adjacent is they share at least one vertex. Mahler and Srecko play a game on this board. Initially, Mahler has one piece placed on the square marked M, and Srecko has a piece placed on the square marked by S (see the attachment). The players alternate moving their piece, following three rules: 1. A piece can only be moved to a unit square adjacent to the one it is placed on. 2. A piece cannot be placed on a unit square on which a piece has been placed before (once used, a unit square can never be used again). 3. A piece cannot be moved to a unit square adjacent to the square occupied by the opponent’s piece. A player wins the game if his piece gets to the corner diagonally opposite to its starting position (i.e. Srecko moves to $s_p$, Mahler moves to $m_p$) or if the opponent has to move but has no legal move. Mahler moves first. Which player has a winning strategy?