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A good looking inequality

Source: European Mathematical Cup 2013, Junior Division, P4

July 3, 2014
Inequalitythree variable inequality

Problem Statement

Let a,b,ca,b,c be positive reals satisfying : a1+b+c+b1+c+a+c1+a+b≥ab1+a+b+bc1+b+c+ca1+c+a \frac{a}{1+b+c}+\frac{b}{1+c+a}+\frac{c}{1+a+b}\ge \frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ca}{1+c+a} Then prove that : a2+b2+c2ab+bc+ca+a+b+c+2≥2(ab+bc+ca) \frac{a^2+b^2+c^2}{ab+bc+ca}+a+b+c+2\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})
Proposed by Dimitar Trenevski
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