For $n=3$, the idea is simple. So we have three numbers: $(A, G, H)$ and three unknowns $(a_1, a_2, a_3)$ such that $A$ is the AM of these numbers, $G$ is the GM, and $H$ is the HM. So we can write a cubic equation such that $a_1, a_2, a_3$ are the roots. This cubic turns out to be
$$
x^3 - 3A x^2 + \frac{3G^3}{H} x - G^3 = 0.
$$
Now write out the condition that this cubic has 3 real, nonnegative roots...