CGMO 2006 Problem 7

Given that $x_{i}>0$ , $i = 1, 2, \cdots, n$ , $k \geq 1$ . Show that: $$\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}$$