## Maximising a weighted product

### Problem 190

Published on Friday, 18th April 2008, 10:00 pm; Solved by 2689; Difficulty rating: 50%Let S_{m} = (x_{1}, x_{2}, ... , x_{m}) be the m-tuple of positive real numbers with x_{1} + x_{2} + ... + x_{m} = m for which P_{m} = x_{1} * x_{2}^{2} * ... * x_{m}^{m} is maximised.

For example, it can be verified that [P_{10}] = 4112 ([ ] is the integer part function).

Find Σ[P_{m}] for 2 ≤ m ≤ 15.