ÖMO 1987 Problem 3

Let $x_1,...,x_n$ be positive real numbers. Prove that:

$\sum_{k=1}^{n}x_k+\sqrt{ \sum_{k=1}^{n}x_k^2} \le \frac{n+\sqrt{n}}{n^2} \left( \sum_{k=1}^{n} \frac{1}{x_k} \right) \left( \sum_{k=1}^{n} x_k^2 \right).$