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Integer inequality with integers on a circle

Source: German TSTST 2016 - #2

July 9, 2016
inequalitiesIntegernumber theorynumber theory unsolvedn-variable inequality

Problem Statement

The positive integers a1,a2,…,ana_1,a_2, \dots, a_n are aligned clockwise in a circular line with n≥5n \geq 5. Let a0=ana_0=a_n and an+1=a1a_{n+1}=a_1. For each i∈{1,2,…,n}i \in \{1,2,\dots,n \} the quotient qi=ai−1+ai+1ai q_i=\frac{a_{i-1}+a_{i+1}}{a_i} is an integer. Prove 2n≤q1+q2+⋯+qn<3n. 2n \leq q_1+q_2+\dots+q_n < 3n.
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