CGMO 2018 Problem 3

Given a real sequence $\left \{ x_n \right \}_{n=1}^{\infty}$ with $x_1^2 = 1$ . Prove that for each integer $n \ge 2$ , $$\sum_{i|n}\sum_{j|n}\frac{x_ix_j}{\textup{lcm} \left ( i,j \right )} \ge \prod_{\text{\tiny $\begin{array}{c} p \: \textup{is prime} p|n \end{array}$ } }\left ( 1-\frac{1}{p} \right ).$$