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a,b,c>0 , abc=1 => Π (ab + bc +1/ca)>=Π(1+2a)

Source: JBMO 2008 Shortlist A7

October 14, 2017
JBMOinequalitiesalgebra

Problem Statement

Let a,ba, b and cc be positive real numbers such that abc=1abc = 1. Prove the inequality (ab+bc+1ca)(bc+ca+1ab)(ca+ab+1bc)≥(1+2a)(1+2b)(1+2c)\Big(ab + bc +\frac{1}{ca}\Big)\Big(bc + ca +\frac{1}{ab}\Big)\Big(ca + ab +\frac{1}{bc}\Big)\ge (1 + 2a)(1 + 2b)(1 + 2c).
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