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The largest constant K with given propery and 4 real numbers

Source: Serbian National Olympiad 2013, Problem 6

April 8, 2013
inequalitiesinequalities proposed

Problem Statement

Find the largest constant K∈RK\in \mathbb{R} with the following property: if a1,a2,a3,a4>0a_1,a_2,a_3,a_4>0 are numbers satisfying ai2+aj2+ak2≥2(aiaj+ajak+akai)a_i^2 + a_j^2 + a_k^2 \geq 2 (a_ia_j + a_ja_k + a_ka_i), for every 1≤i<j<k≤41\leq i<j<k\leq 4, then a12+a22+a32+a42≥K(a1a2+a1a3+a1a4+a2a3+a2a4+a3a4).a_1^2+a_2^2+a_3^2+a_4^2 \geq K (a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4).
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