Let a and b be distinct positive integers. The following infinite process takes place on an initially empty board.
[*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by a and the other by b.
[*] If no such pair exists, we write two times the number 0.Prove that, no matter how we make the choices in (i), operation (ii) will be performed only finitely many times.Proposed by [I]Serbia[/I]. IMO ShortlistcombinatoricsIMO shortlist 2018