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3 variable

Source: Middle European Mathematical Olympiad 2013 I-1

May 17, 2014
inequalities proposedinequalities

Problem Statement

Let a,b,c a, b, c be positive real numbers such that a+b+c=1a2+1b2+1c2. a+b+c=\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} . Prove that 2(a+b+c)≥7a2b+13+7b2c+13+7c2a+13. 2(a+b+c) \ge \sqrt[3]{7 a^2 b +1 } + \sqrt[3]{7 b^2 c +1 } + \sqrt[3]{7 c^2 a +1 } . Find all triples (a,b,c) (a,b,c) for which equality holds.
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