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Non-negative solutions of inequality

Source: Czech and Slovak Olympiad 1990, National Round, Problem 4

October 12, 2024
inequalitiesalgebramaxparameter

Problem Statement

Determine the largest k≥0k\ge0 such that for every n≥2n\ge2 and any nn-tuple x1,…,xnx_1,\ldots,x_n of non-negative numbers is a solution of the inequality (given that xn+1=x1x_{n+1}=x_1) (∑j=1nxj)2(∑j=1nxjxj+1)≥k∑j=1nxj2xj+12.\left(\sum_{j=1}^n x_j\right)^2\left(\sum_{j=1}^n x_jx_{j+1}\right)\ge k\sum_{j=1}^n x_j^2x_{j+1}^2.
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