8.5. Miron and Varya are playing the following game. There is a pile of 32 stones on the table. The players take turns, with Miron starting first. On their turn, a player divides any pile that has more than one stone into several equal piles. The player who cannot make a move (when there is exactly one stone in each pile before their turn) wins. Who will win if both players play optimally?
Answer: Varya.

Solution. Varya wins regardless of the moves the players make. Since the number 32 has no odd divisor, each player will divide one pile into an even number of piles on their turn, and the parity of the number of piles will change. Before Miron's move, there will always be an odd number of piles, and before Varya's move, there will be an even number. Since at the end there will be 32 unit piles in front of the winner, the winner is Varya.

Comment. An answer without justification - 0 points.