KöMaL 2009 B4217

Let $a$, $b$, $c$ denote the sides of an acute triangle and let $\gamma$, $\beta$, $\alpha$, respectively, denote the opposite angles (in radians). Show that $\frac{\pi}{3}\le \frac{\alpha a^2+\beta b^2+\gamma c^2}{a^2+b^2+c^2}<\frac{\pi}{2}$.