5

Part of 1983 Canada National Olympiad

Problems(1)

GM of set = GM of the GMs of all non-zero subsets

Source: Canadian Mathematical Olympiad - 1983 - Problem 5.

The geometric mean (G.M.) of

k

positive integers

S

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

n

positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of

S

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