IMO 1987 LL USA64

Origin: USA

Problem

Let r > 1 be a real number, and let n be the largest integer smaller than r. Consider an arbitrary real number x with 0 \leqx \leq n r−1. By a base-r expansion of x we mean a representation of x in the form x = a1 r + a2 r2 + a3 r3 + \cdot \cdot \cdot , where the ai are integers with 0 \leqai < r. You may assume without proof that every number x with 0 \leqx \leq n r−1 has at least one base-r expansion. Prove that if r is not an integer, then there exists a number p, 0 \leqp \leq n r−1, which has infinitely many distinct base-r expansions.