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2015 German National Olympiad, 6
Tags: mean , harmonic mean , inequalities
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.