CMO 1983 Problem 5

The geometric mean (G.M.) of $k$ positive integers $a_1$ , $a_2$ , $\dots$ , $a_k$ is defined to be the (positive) $k$ -th root of their product. For example, the G.M. of 3, 4, 18 is 6. Show that the G.M. of a set $S$ of $n$ positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of $S$ .