KöMaL 2012 B4442

Ann, Bill and Cecilia play the following game. They take turns choosing an integer between 1 and 10, and adding that number to the sum of the numbers that have been said so far. (For simplicity, in each turn, the player states the sum.) The player who is first able to say 100 wins the game. Prove that the two girls, with the appropriate strategy, are able to achieve that one of them is the winner.