RU-ALLRUSSIAN 2018 Problem 112

Let $n\geq 2$ and $x_{1},x_{2},\ldots,x_{n}$ positive real numbers. Prove that $$\frac{1+x_{1}^2}{1+x_{1}x_{2}}+\frac{1+x_{2}^2}{1+x_{2}x_{3}}+\cdots+\frac{1+x_{n}^2}{1+x_{n}x_{1}}\geq n.$$