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Problem 1 -- Constant Inequality Chain

Source: 47th Austrian Mathematical Olympiad National Competition Part 1 Problem 1

July 28, 2018
inequalitiesAustriaalgebra

Problem Statement

Determine the largest constant CC such that
(x1+x2+⋯+x6)2≥C⋅(x1(x2+x3)+x2(x3+x4)+⋯+x6(x1+x2))(x_1 + x_2 + \cdots + x_6)^2 \ge C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \cdots + x_6(x_1 + x_2))
holds for all real numbers x1,x2,⋯ ,x6x_1, x_2, \cdots , x_6.
For this CC, determine all x1,x2,⋯x6x_1, x_2, \cdots x_6 such that equality holds.
(Walther Janous)
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