IMO 2013 Shortlist A4
Let n be a positive integer, and consider a sequence a1,a2,...,an of positive integers. Extend it periodically to an inf...
Category: Algebra
Problem
Let n be a positive integer, and consider a sequence a1,a2,...,an of positive integers.
Extend it periodically to an infinite sequence a1,a2,... by defining ani “ ai for all i ě 1. If a1 ď a2 ď ¨¨¨ ď an ď a1 n
and
aai
ď n i ´ 1 for i “ 1,2,...,n, prove that a1 ¨¨¨ ` an ď n2
.
(Germany)