IMO 2024 Shortlist A6
Let a0, a1, a2, ... be an infinite strictly increasing sequence of positive integers such that for each n ě 1 we have an...
Category: Algebra
Problem
Let a0, a1, a2, ... be an infinite strictly increasing sequence of positive integers such
that for each n ě 1 we have
an P
!an´1 an1
,
?
an´1 ¨ an`1
)
.
Let b1, b2, ... be an infinite sequence of letters defined as
bn “
A, if an “ 1
pan´1 an1q;
G, otherwise.
Prove that there exist positive integers n0 and d such that for all n ě n0 we have bn`d “ bn.
(Czech Republic)