DE-MO 2015 Problem 6

Prove that for all $x,y,z>0$ , the inequality $$\frac{x+y+z}{3}+\frac{3}{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}} \ge 5 \sqrt[3]{\frac{xyz}{16}}$$ holds. Determine if equality can hold and if so, in which cases it occurs.