Q2 Matemática  (Mediterranean Mathematics Olympiad 2012)

Em um $△ABC$ agudo, prove que $$\begin{array}{r}\frac{1}{3}\left(\frac{{\mathrm{tg}}^{2}A}{\mathrm{tg}B\mathrm{tg}C}+\frac{{\mathrm{tg}}^{2}B}{\mathrm{tg}C\mathrm{tg}A}+\frac{{\mathrm{tg}}^{2}C}{\mathrm{tg}A\mathrm{tg}B}\right)\\ +3{\left(\frac{1}{\mathrm{tg}A+tanB+\mathrm{tg}C}\right)}^{\frac{2}{3}}\ge 2.\end{array}$$