KöMaL 2014 B4638

Let $x_{1}, x_{2}, \ldots, x_{n}$ denote arbitrary real numbers. Prove that $\sqrt{\left(\sum_{k=1}^{n} \frac{x_{k}^{4}+k^{2}}{x_{k}^{2}}\right)^{2}-n^{2} (n+1)^{2}}\ge \sum_{k=1}^{n} \frac{x_{k}^{4}-k^{2}}{x_{k}^{2}}$.