Question 21
Let $a$ and $b$ be distinct positive integers. There is an infinite chessboard, each cell of which can be filled with a nonnegative integer. The board is initially empty. We repeatedly perform the following operations:
(i) If a pair of equal numbers can be found on the board, then we pick two cells containing the same number, increase the number in one of them by $a$ and the number in the other by $b$.
(ii) If no pair of equal numbers can be found on the board, then we fill a 0 into each of two empty cells.
Prove that, no matter what, from some operation on only operation (i) can be performed.
(i) If a pair of equal numbers can be found on the board, then we pick two cells containing the same number, increase the number in one of them by $a$ and the number in the other by $b$.
(ii) If no pair of equal numbers can be found on the board, then we fill a 0 into each of two empty cells.
Prove that, no matter what, from some operation on only operation (i) can be performed.