# [Solutions] Balkan Mathematical Olympiad 2021

1. Let $ABC$ be a triangle with $AB<AC$. Let $\omega$ be a circle passing through $B$, $C$ and assume that $A$ is inside $\omega$. Suppose $X$, $Y$ lie on $\omega$ such that $\angle BXA=\angle AYC$. Suppose also that $X$ and $C$ lie on opposite sides of the line $AB$ and that $Y$ and $B$ lie on opposite sides of the line $AC$. Show that, as $X$, $Y$ vary on $\omega$, the line $XY$ passes through a fixed point.
2. Find all functions $f:\mathbb R_+ \to \mathbb R_+$, such that $$f(x+f(x)+f(y))=2f(x)+y$$ for all positive reals $x,y$.
3. 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.
4. 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
• He clears every piece of rubbish from a single pile.
• 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
• He adds one piece of rubbish to each non-empty pile.
• 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?
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