IMO 2015 Shortlist N4
Suppose that a0,a1,... and b0,b1,... are two sequences of positive integers satisfying a0,b0 ě 2 and an1 “ gcdpan,bnq 1,...
Category: Number Theory
Problem
Suppose that a0,a1,... and b0,b1,... are two sequences of positive integers satisfying
a0,b0 ě 2 and
an1 “ gcdpan,bnq 1, bn1 “ lcmpan,bnq ´ 1 for all n ě 0. Prove that the sequence (an) is eventually periodic; in other words, there exist integers N ě 0 and t ą 0 such that ant “ an for all n ě N.
(France)