[60] - Algebraic Inequalities from and for Math Contests

Έστω x,y,z >0. Να αποδειχθεί ότι: $$\dfrac{x+y+z}{3}+\dfrac{3}{\dfrac1x+\dfrac1y+\dfrac1z}\geq5\sqrt[3]{\dfrac{xyz}{16}}.$$