Problem

Source: Germany 2015, Problem 6

Tags: inequalities, inequalities proposed, mean, Harmonic Mean



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.