Balkan Mathematical Olympiad 2023 [Solutions]

1. Find all functions $f:\mathbb{R}\to \mathbb{R}$ 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 $\mathrm{\angle }XBC=\mathrm{\angle }XCB={45}^{\circ }$. 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 $k\le 2023$ 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.

DOWNLOAD SOLUTIONS