[ Combinatorial geometry (other).] [ Symmetric strategy ]

Kolya and Vitya are playing the following game on an infinite grid paper. Starting with Kolya, they take turns marking the nodes of the grid paper - the points of intersection of vertical and horizontal lines. Each of them, on their turn, must mark such a node that after this, all marked nodes lie at the vertices of a convex polygon (starting from Kolya's second move). The player who cannot make the next move is considered the loser. Who wins with correct play?
Answer: Vitya. After Kolya's first move, Vitya mentally marks an arbitrary node $O$, different from the one marked by Kolya. Then he marks a node that is symmetric to $O$ relative to the node marked by Kolya each time. It is clear that this again results in a convex polygon. After six moves, a centrally symmetric hexagon is formed. In further moves, only nodes lying in the six triangles formed by the sides of the hexagon and the extensions of the sides can be marked. Therefore, Kolya has only a finite number of possible moves.