CZ-SK-MO 2023

Alice and Bob are playing a game on a plane consisting of $72$ cells arranged in circle. At the beginning of the game, Bob places a stone on some of the cells. Then, in every round first Alice picks one empty cell and then Bob must move a stone from one of the two neighboring cells on this cell. If he is unable to do that, game ends. Determine the smallest number of stones he has to place in the beginning so he has a strategy to make the game last for at least $2023$ rounds.

Let $n$ be a positive integer, where $n \geq 3$ and let $a_1, a_2, ..., a_n$ be the lengths of sides of some $n$ -gon. Prove that $$a_1 + a_2 + ... + a_n \geq \sqrt{2 \cdot (a_1^2 + a_2^2 + ... + a_n^2)}$$

In acute triangle $ABC$ let $H$ be its orthocenter and $I$ be its incenter. Let $D$ be the projection of point $I$ onto the line $BC$ and $E$ be the reflection of point $A$ in point $I$ . Further, let $F$ be the projection of point $H$ onto the line $ED$ . Prove that points $B, H, F$ and $C$ lie on circle.

In triangle $ABC$ let $N, M, P$ be the midpoints of the sides $BC, CA, AB$ and $G$ be the centroid of this triangle. Let the circle circumscribed to $BGP$ intersect the line $MP$ in point $K$ , $P \neq K$ , and the circle circumscribed to $CGN$ intersect the line $MN$ in point $L$ , $N \neq L$ . Prove that $\angle BAK = \angle CAL$ .