Problem 190
Maximising a weighted product

Let Sm = (x1, x2, ... , xm) be the m-tuple of positive real numbers with x1 + x2 + ... + xm = m for which Pm = x1 * x22 * ... * xmm is maximised.

For example, it can be verified that [P10] = 4112 ([ ] is the integer part function).

Find Σ[Pm] for 2 ≤ m ≤ 15.

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http://projecteuler.net/problem=190