Question 175
Let $n \ge 3$ be an integer and $x_1, x_2, \dots, x_{n-1}$ be nonnegative integers such that
(i) $x_1 + x_2 + \dots + x_{n-1} = n$,
(ii) $x_1 + 2x_2 + \dots + (n-1)x_{n-1} = 2n - 2$.
Find the minimum of the sum $\sum_{k=1}^{n-1} kx_k(2n-k)$. Justify your answer.
(i) $x_1 + x_2 + \dots + x_{n-1} = n$,
(ii) $x_1 + 2x_2 + \dots + (n-1)x_{n-1} = 2n - 2$.
Find the minimum of the sum $\sum_{k=1}^{n-1} kx_k(2n-k)$. Justify your answer.