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Maximising a Weighted Product

 Published on Friday, 18th April 2008, 10:00 pm; Solved by 4407;
Difficulty rating: 50%

Problem 190

Let $S_m = (x_1, x_2, \dots , x_m)$ be the $m$-tuple of positive real numbers with $x_1 + x_2 + \cdots + x_m = m$ for which $P_m = x_1 \cdot x_2^2 \cdot \cdots \cdot x_m^m$ is maximised.

For example, it can be verified that $\lfloor P_{10}\rfloor = 4112$ ($\lfloor \, \rfloor$ is the integer part function).

Find $\sum\limits_{m = 2}^{15} \lfloor P_m \rfloor$.