# [Solutions] Balkan Mathematical Olympiad 2022

1. Let $ABC$ be an acute triangle such that $CA \neq CB$ with circumcircle $\omega$ and circumcentre $O$. Let $t_A$ and $t_B$ be the tangents to $\omega$ at $A$ and $B$ respectively, which meet at $X$. Let $Y$ be the foot of the perpendicular from $O$ onto the line segment $CX$. The line through $C$ parallel to line $AB$ meets $t_A$ at $Z$. Prove that the line $YZ$ passes through the midpoint of the line segment $AC$.
2. Let $a, b$ and $n$ be positive integers with $a>b$ such that all of the following hold
• $a^{2021}$ divides $n$,
• $b^{2021}$ divides $n$,
• $2022$ divides $a-b$.
Prove that there is a subset $T$ of the set of positive divisors of the number $n$ such that the sum of the elements of $T$ is divisible by $2022$ but not divisible by $2022^2$.
3. Find all functions $f: (0, \infty) \to (0, \infty)$ such that $$f(y(f(x))^3 + x) = x^3f(y) + f(x),\,\forall x, y>0.$$
4. Consider an $n \times n$ grid consisting of $n^2$ until cells, where $n \geq 3$ is a given odd positive integer. First, Dionysus colours each cell either red or blue. It is known that a frog can hop from one cell to another if and only if these cells have the same colour and share at least one vertex. Then, Xanthias views the colouring and next places $k$ frogs on the cells so that each of the $n^2$ cells can be reached by a frog in a finite number (possible zero) of hops. Find the least value of $k$ for which this is always possible regardless of the colouring chosen by Dionysus.
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