ISL 1991 Problem 30

Two students $A$ and $B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $A:$ "Can you tell the integer written by the other student?" If A answers "no," the referee puts the same question to student $B.$ If $B$ answers "no," the referee puts the question back to $A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer "yes."