13 - Inequalities

Let $a, b, c$ be positive real numbers such that 

$\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} ≥ 1$.

Prove that

$\dfrac{a + b}{\sqrt{ab + c}} + \dfrac{b+c}{\sqrt{bc + a}} + \dfrac{c+a}{\sqrt{ca + b}} ≥ 3\sqrt[6]{abc}$.

Proposed by Titu Andreescu, University of Texas at Dallas, USA

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