Let $ABC$ be a scalene triangle with circumcircle $\omega$. The tangent to $\omega$ at $A$ meets $BC$ at $D$. The $A$-median of triangle $ABC$ intersects $BC$ and $\omega$ at $M$ and $N$, respectively. Suppose that $K$ is a point such that $ADMK$ is a parallelogram. Prove that $KA = KN$. Proposed by Alexandru Lopotenco (Moldova)

Consider some graph $G$ with $2019$ nodes. Let's define inverting a vertex $v$ the following process: for every other vertex $u$, if there was an edge between $v$ and $u$, it is deleted, and if there wasn't, it is added. We want to minimize the number of edges in the graph by several invertings (we are allowed to invert the same vertex several times). Find the smallest number $M$ such that we can always make the number of edges in the graph not larger than $M$, for any initial choice of $G$. Proposed by Arsenii Nikolaev, Anton Trygub (Ukraine)

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all real $x, y$, the following relation holds: $$(x+y) \cdot f(x+y)= f(f(x)+y) \cdot f(x+f(y)).$$ Proposed by Vadym Koval (Ukraine)

Call a two-element subset of $\mathbb{N}$ cute if it contains exactly one prime number and one composite number. Determine all polynomials $f \in \mathbb{Z}[x]$ such that for every cute subset $ \{ p,q \}$, the subset $ \{ f(p) + q, f(q) + p \} $ is cute as well. Proposed by Valentio Iverson (Indonesia)

Find all pairs of positive integers $(s, t)$, so that for any two different positive integers $a$ and $b$ there exists some positive integer $n$, for which $$a^s + b^t | a^n + b^{n+1}.$$ Proposed by Oleksii Masalitin (Ukraine)

Let $ABC$ be a scalene triangle with incenter $I$ and circumcircle $\omega$. The internal and external bisectors of angle $\angle BAC$ intersect $BC$ at $D$ and $E$, respectively. Let $M$ be the point on segment $AC$ such that $MC = MB$. The tangent to $\omega$ at $B$ meets $MD$ at $S$. The circumcircles of triangles $ADE$ and $BIC$ intersect each other at $P$ and $Q$. If $AS$ meets $\omega$ at a point $K$ other than $A$, prove that $K$ lies on $PQ$. Proposed by Alexandru Lopotenco (Moldova)