5. The numbers $1,2,3, \ldots, 99$ are written on the board. Petya and Vasya are playing a game, with Petya starting. Each move involves erasing three numbers that sum to 150. The player who cannot make a move loses. Which player can win, regardless of how the opponent plays?
# 5. Answer: Petya.

Solution: His first move is to erase the number 50 and any 2 numbers that sum to 100, for example, 1 and 99. He then divides all the remaining numbers into pairs with a sum of 100: (2,98), (3,97), ..., (49,51).

Vasya cannot erase two numbers from the same pair in one move, as their sum is 100, and he is short of the number 50 to reach 150.

Therefore, Vasya erases 3 numbers from different pairs, and Petya erases the remaining numbers from the same pairs.

Since the sum of the numbers in three pairs is 300, and Vasya has erased three numbers with a sum of 150, Petya does the same.

Thus, Petya always has a response to Vasya's move.

## Grading Instructions:

Award 3 points for the correct strategy without justification.