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IMO Shortlist 2011, Algebra 7

Source: IMO Shortlist 2011, Algebra 7

July 11, 2012
inequalitiesalgebraIMO Shortlist

Problem Statement

Let a,ba,b and cc be positive real numbers satisfying min⁡(a+b,b+c,c+a)>2\min(a+b,b+c,c+a) > \sqrt{2} and a2+b2+c2=3.a^2+b^2+c^2=3. Prove that
a(b+c−a)2+b(c+a−b)2+c(a+b−c)2≥3(abc)2.\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.
Proposed by Titu Andreescu, Saudi Arabia
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