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Problem 2 of Fourth round

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

December 31, 2019
number theorySequence

Problem Statement

Let a0,a1,a2...a_0,a_1,a_2... be a sequence of natural numbers with the following property: an2a_n^2 divides an−1an+1a_{n-1} a_{n+1} for ∀\forall n∈Nn\in \mathbb{N}. Prove that, if for some natural k≥2k\geq 2 the numbers a1a_1 and aka_k are coprime, then a1a_1 divides a0a_0.
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