BALKAN-SL 2018 A6

Let $n$ be a positive integer and let $x_{1}, \ldots, x_{n}$ be real numbers. Show that

$$\sum_{i=1}^{n} x_{i}^{2} \geqslant \frac{1}{n+1}\left(\sum_{i=1}^{n} x_{i}\right)^{2}+\frac{12\left(\sum_{i=1}^{n} i x_{i}\right)^{2}}{n(n+1)(n+2)(3 n+1)}$$