KöMaL 2016 B4828

Let $0<x_1<x_2<\ldots<x_n<2\pi$. Show that $\sum_{i,j=1; i\ne j}^n \frac{1}{|x_i-x_j|}+\frac{1}{2\pi-|x_i-x_j|}\ge \frac{n^2}{\pi}\sum_{k=1}^{n-1} \frac 1k.$ On what condition will the equality hold?