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Find the minimal value of F(x_1,x_2,\ldots,x_n)

Source: Romanian IMO Team Selection Test TST 1996, problem 9

September 27, 2005
inequalities proposedinequalities

Problem Statement

Let n≥3 n\geq 3 be an integer and let x1,x2,…,xn−1 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(x1,x2,…,xn)=∑k=1n−1k(2n−k)xk F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k .
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