Darij Grinberg's Inequality In Three Variables

Problem

Darij Grinberg's Inequality In Three Variables

Solution

Of the three numbers $1-a,\,$ $1-b,\,$ $1-c,\,$ at least two are of the same sign. WLOG, let's say $(1-b)(1-c)\ge 0.\,$ Then

$\begin{align} &a^2+b^2+c^2+2abc+1-2(ab+bc+ca)\\ &=(a-1)^2+(b-c)^2+2a+2abc-2(ab+ca)\\ &=(a-1)^2+(b-c)^2+2a(1-b)(1-c)\ge 0. \end{align}$

Equality occurs for $a=b=c=1.$

Acknowledgment

This is Problem 5 from Vasile Cîrtoaje book Algebraic Inequalities: Old and New Methods. The problem is by Darij Grinberg (2004), the above solution is probably by Vasile Cîrtoaje.

 

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