Balkan Mathematical Olympiad 2023 [Solutions]

1. Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, \[xf(x+f(y))=(y-x)f(f(x)).\]

2. In triangle $ABC$, the incircle touches sides $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Assume there exists a point $X$ on the line $EF$ such that $\angle{XBC} = \angle{XCB} = 45^{\circ}$. Let $M$ be the midpoint of the arc $BC$ on the circumcircle of $ABC$ not containing $A$. Prove that the line $MD$ passes through $E$ or $F$. 

3. For each positive integer $n$, denote by $\omega(n)$ the number of distinct prime divisors of $n$ (for example, $\omega(1)=0$ and $\omega(12)=2$). Find all polynomials $P(x)$ with integer coefficients, such that whenever $n$ is a positive integer satisfying $\omega(n)>2023^{2023}$, then $P(n)$ is also a positive integer with \[\omega(n)\ge\omega(P(n)).\]4. Find the greatest integer $k\leq 2023$ for which the following holds: whenever Alice colours exactly $k$ numbers of the set $\{1,2,\dots, 2023\}$ in red, Bob can colour some of the remaining uncoloured numbers in blue, such that the sum of the red numbers is the same as the sum of the blue numbers.

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