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Kazakhstan National Olympiad 2017, Final Round, 11-Grade, P3

Source: Kazakhstan National Olympiad 2017, Final Round, 11-Grade, P3

March 18, 2017

Sequencealgebra

Problem Statement

\{a_n\}

is an infinite, strictly increasing sequence of positive integers and

a_{a_n}\leq a_n+a_{n+3}

for all

n\geq 1

. Prove that, there are infinitely many triples

(k,l,m)

of positive integers such that

k<l<m

and

a_k+a_m=2a_l