BALKAN-SL 2008 A7

Let $x,y,z,t \in \mathbb{R}_{\geq 0}$ . Show \begin{align*} \sqrt{xy}+\sqrt{xz}+\sqrt{xt}+\sqrt{yz}+\sqrt{yt}+\sqrt{zt} \geq 3 \sqrt[3]{xyz+xyt+xzt+yzt} \end{align*} and determine the equality cases.