BALKAN 2021

Let $A B C$ be a triangle with $A B<A C$. Let $\omega$ be a circle passing through $B, C$ and assume that $A$ is inside $\omega$. Suppose $X, Y$ lie on $\omega$ such that $\angle B X A=\angle A Y C$. Suppose also that $X$ and $C$ lie on opposite sides of the line $A B$ and that $Y$ and $B$ lie on opposite sides of the line $A C$.

Show that, as $X, Y$ vary on $\omega$, the line $X Y$ passes through a fixed point.

Find all functions $f:(0,+\infty) \rightarrow(0,+\infty)$ such that

$$f(x+f(x)+f(y))=2 f(x)+y$$

holds for all $x, y \in(0,+\infty)$.

Let $a, b$ and $c$ be positive integers satisfying the equation

$$(a, b)+[a, b]=2021^{c} .$$

If $|a-b|$ is a prime number, prove that the number $(a+b)^{2}+4$ is composite. Here, $(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $[a, b]$ denotes the least common multiple of $a$ and $b$.

Angel has a warehouse, which initially contains 100 piles of 100 pieces of rubbish each. Each morning, Angel performs exactly one of the following moves: (a) He clears every piece of rubbish from a single pile. (b) He clears one piece of rubbish from each pile.

However, every evening, a demon sneaks into the warehouse and performs exactly one of the following moves: (a) He adds one piece of rubbish to each non-empty pile. (b) He creates a new pile with one piece of rubbish.

What is the first morning when Angel can guarantee to have cleared all the rubbish from the warehouse?