IMO 1988 LL SWE80

Origin: SWE

Problem

Let S be an infinite set of integers containing zero and such that the distance between successive numbers never exceeds a given fixed number. Consider the following procedure: Given a set X of integers, we construct a new set consisting of all numbers x \pm s, where x belongs to X and s belongs to S. Starting from S0 = {0} we successively construct sets S1, S2, S3, . . . using this procedure. Show that after a finite number of steps we do not obtain any new sets; i.e., Sk = Sk0 for k \geqk0.