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cyc sum (a+1)\sqrt{2a(1-a)} \geq 8(ab+bc+ca)

Source: Greece JBMO TST 2017, Problem 1

June 25, 2018
inequalities

Problem Statement

Positive real numbers a,b,ca,b,c satisfy a+b+c=1a+b+c=1. Prove that (a+1)2a(1−a)+(b+1)2b(1−b)+(c+1)2c(1−c)≥8(ab+bc+ca).(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca). Also, find the values of a,b,ca,b,c for which the equality happens.
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