BALKAN-SL 2008 A1

For all $\alpha_1, \alpha_2,\alpha_3 \in \mathbb{R}^+$ , Prove \begin{align*} \sum \frac{1}{2\nu \alpha_1 +\alpha_2+\alpha_3} > \frac{2\nu}{2\nu +1} \left( \sum \frac{1}{\nu \alpha_1 + \nu \alpha_2 + \alpha_3} \right) \end{align*} for every positive real number $\nu$