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Inequality with odd numbers.

Source: EGMO 2016 Day 1 Problem 1

April 12, 2016
InequalityalgebrainequalitiesEGMOn-variable inequalitySequence

Problem Statement

Let nn be an odd positive integer, and let x1,x2,⋯ ,xnx_1,x_2,\cdots ,x_n be non-negative real numbers. Show that min⁡i=1,…,n(xi2+xi+12)≤max⁡j=1,…,n(2xjxj+1) \min_{i=1,\ldots,n} (x_i^2+x_{i+1}^2) \leq \max_{j=1,\ldots,n} (2x_jx_{j+1}) where xn+1=x1x_{n+1}=x_1.
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