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An inequality with condition sum a/(1+a)=2

Source: Middle European Mathematical Olympiad 2011 - Team Compt. T-2

September 6, 2011
inequalitiesCauchy Inequalityinequalities unsolved

Problem Statement

Let a,b,ca, b, c be positive real numbers such that a1+a+b1+b+c1+c=2.\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=2. Prove that a+b+c2≥1a+1b+1c.\frac{\sqrt a + \sqrt b+\sqrt c}{2} \geq \frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}.
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