Let a_{1} < a_{2} < a_{3} < ... be positive integers. Prove that for every i >= 1, there are infinitely many a_{n} that can be written in the form a_{n} = ra_{i} + sa_{j}, with r, s positive integers and j > i.

**Solution**

We must be able to find a set S of infinitely many a_{n} in some residue class mod a_{i}. Take a_{j} to be a member of S. Then for any a_{n} in S satisfying a_{n} > a_{j}, we have a_{n} = a_{j} + a multiple of a_{i}.

Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.

© John Scholes

jscholes@kalva.demon.co.uk

10 Oct 1998