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

The best solution was submitted by Sang Hoon Kwon (권상훈), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2008-10.