IMO 1975

Let a₁ < a₂ < a₃ < ... 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.

17th IMO 1975

© John Scholes
jscholes@kalva.demon.co.uk
10 Oct 1998