55rd Ukrainian National Mathematical Olympiad - Third Round

Question

2015 candies are placed along a circle and numbered 1 to 2015 clockwise. Andriy and Olesia play the following game. In each turn, a player can take either 2 or 3 candies with consecutive numbers (1 and 2015 are also considered "consecutive"). The player who can't make a move loses. Who has a winning strategy if Andriy plays first?

Accepted Answer

Andriy takes 2 (or 3) consecutive candies. Then Olesia takes 3 (or 2) diametrically opposite candies, so that there are 1005 candies on both sides between the groups of taken candies. Then she just copies Andriy's moves on the other part. If Andriy can make a move, she can as well, so she will not lose anyway. Since the game is finite, in the end she will win.