Maximising a weighted product
Problem 190Published on Friday, 18th April 2008, 10:00 pm; Solved by 2651; Difficulty rating: 50%
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.