2009 IMO Problems/Problem 3

Problem

Suppose that $s_1,s_2,s_3,\ldots$ is a strictly increasing sequence of positive integers such that the subsequences

$s_{s_1},s_{s_2},s_{s_3},\ldots$ and $s_{s_1+1},s_{s_2+1},s_{s_3+1},\ldots$

are both arithmetic progressions. Prove that the sequence $s_1,s_2,s_3,\ldots$ is itself an arithmetic progression.

Author: Gabriel Carroll, USA

--Bugi 10:22, 23 July 2009 (UTC) Bugi