BY-MO 1995 Problem 7

The expression $1\oplus2\oplus3\oplus4\oplus5\oplus6\oplus7\oplus8\oplus9$ is written on a blackboard. Bill and Peter play the following game. They replace $\oplus$ by $+$ or $\cdot$ , making their moves in turn, and one of them can use only $+$ , while the other one can use only $\cdot$ . At the beginning, Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win.