MathNet 2010 — Part 04 Question 193

Given an integer number $n \ge 3$, determine the real numbers $x_1, x_2, \dots, x_n$ minimising $(n-1)(x_1^2 + x_2^2 + \dots + x_n^2) + nx_1x_2 \dots x_n$, subject to $x_k \ge 0$, $k = 1, 2, \dots, n$, and $x_1 + x_2 + \dots + x_n = n$.