This was asked by my professor.

Determine which positive integers $n$ have the following property: If $a[1] , dots, a[n]$ are $n$ real numbers greater than or equal to 1, and $A$, $G$, and $H$ are their arithmetic mean, geometric mean, and harmonic mean, respectively, then: $$G-H ≥ frac 1G – frac 1A$$