# 2008-10 Inequality with n variables

Let $$x_1,x_2,\ldots,x_n$$ be nonnegative real numbers. Show that
$$\displaystyle \left(\sum_{i=1}^n x_i\right) \left(\sum_{i=1}^n x_i^{n-1}\right) \le (n-1) \sum_{i=1}^n x_i^n + n \prod_{i=1}^n x_i$$.

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