KöMaL 2012 B4478

Prove that if $\alpha$, $\beta$ and $\gamma$ are the angles of an acute-angled triangle then $\mathop{\rm tg}\nolimits^3 \alpha +\mathop{\rm tg}\nolimits^3 \beta +\mathop{\rm tg}\nolimits^3 \gamma \ge 9\sqrt{3}$.