MK-MO 2023

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ we have: $$xf(x+y)+yf(y-x) = f(x^2+y^2)\,.$$ Authored by Nikola Velov

Let $p$ and $q$ be odd prime numbers and $a$ a positive integer so that $p|a^q+1$ and $q|a^p+1$ . Show that $p|a+1$ or $q|a+1$ .

Authored by Nikola Velov

In a city of gnomes there are $1000$ identical towers, each of which has $1000$ stories, with exactly one gnome living on each story. Every gnome in the city wears a hat colored in one of $1000$ possible colors and any two gnomes in the same tower have different hats. A pair of gnomes are friends if they wear hats of the same color, one of them lives in the $k$ -th story of his tower and the other one in the $(k+1)$ -st story of his tower. Determine the maximal possible number of pairs of gnomes which are friends.

Authored by Nikola Velov

Let $ABC$ be a scalene acute triangle with orthocenter $H$ . The circle with center $A$ and radius $AH$ meets the circumcircle of $BHC$ at $T_{a} \neq H$ . Define $T_{b}$ and $T_{c}$ similarly. Show that $H$ lies on the circumcircle of $T_{a}T_{b}T_{c}$ .

Authored by Nikola Velov

There are $n$ boys and $n$ girls sitting around a circular table, where $n>3$ . In every move, we are allowed to swap the places of $2$ adjacent children. The entropy of a configuration is the minimal number of moves such that at the end of them each child has at least one neighbor of the same gender.

Find the maximal possible entropy over the set of all configurations.

Authored by Viktor Simjanoski