Ivenov V.

a) The vertices of a regular 10-gon are painted alternately black and white. Two players play the following game. Each, in turn, draws a segment connecting vertices of the same color. These segments must not have any common points (even endpoints) with those drawn previously. The player who makes the last move wins. Who wins with correct play: the player starting the game or their partner?

b) The same question for a 12-gon.
a) The second player can apply a symmetric strategy: each time, draw a segment that is centrally symmetric to the segment drawn by the first player. In this case, he will always have a move, so he cannot lose.

b) Here, the starting player can apply a symmetric strategy: first, he connects the opposite vertices, and then acts according to the strategy described in a).

## Answer

a) Partner; b) starting player.