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please help!!!! problem from danube cup romania 2010

Source: Donova Mathmatical Olympiad 2010

February 5, 2011
algebrapolynomialfunctioninequalitiescalculusn-variable inequality

Problem Statement

Let n≥3n\ge3 be a positive integer. Find the real numbers x1≥0,…,xn≥0x_1\ge0,\ldots,x_n\ge 0, with x1+x2+…+xn=nx_1+x_2+\ldots +x_n=n, for which the expression (n−1)(x12+x22+…+xn2)+nx1x2…xn(n-1)(x_1^2+x_2^2+\ldots+x_n^2)+nx_1x_2\ldots x_n takes a minimal value.
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