## Maximum Product

#### Problem

For any positive integer, $k$, let $S_k = \{x_1,x_2,...,x_n\}$ be the set of real numbers for which $x_1 + x_2 + ... + x_n = k$ and $P = x_1 x_2 ... x_n$ is maximised.

For example, when $k = 10$, the set $\{2, 3, 5\}$ would give $P = 30$ and the set $\{2.2, 2.4, 2.5, 2.9\}$ would give $P = 38.25$. In fact, $S_10 = \{2.5, 2.5, 2.5, 2.5\}$, for which $P = 39.0625$.

Prove that $P$ is maximised when all the elements of $S$ are equal in value and rational.

Problem ID: 334 (19 Nov 2007) Difficulty: 4 Star