RO-TST 1996 Problem 9

Let $n\geq 3$ be an integer and let $x_1,x_2,\ldots,x_{n-1}$ be nonnegative integers such that

\begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*}

Find the minimal value of $F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k$ .