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Inequality on three variables

Source: Balkan MO ShortList 2008 A6

April 6, 2020

Problem Statement

Prove that if x,y,z∈R+x,y,z \in \mathbb{R}^+ such that xy,yz,zxxy,yz,zx are sidelengths of a triangle and kk ∈\in [−1,1][-1,1], then \begin{align*} \sum \frac{\sqrt{xy}}{\sqrt{xz+yz+kxy}} \geq 2 \sqrt{1-k} \end{align*} Determine the equality condition too.
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