SAUDI ARABIAN MATHEMATICAL COMPETITIONS

We prove by induction on $n$.

When $n=3$, we can assume that $x_1>x_2>x_3$ (only when $n=3$) and the LHS $\ge 1^2+1^2+2^2=6$.

Moving from $n$ to $n+1$, the change in LHS is $$C_{n+1}={\left(x_n-x_{n+1}\right)}^2+{\left(x_{n+1}-x_1\right)}^2-{\left(x_n-x_1\right)}^2.$$ We need to show that $C_{n+1}\ge 4$. We can see that it is not true in general. However, due to the cyclicity, we can assume that $x_{n+1}$ is the least number.

Moreover, since the LHS is unchanged under translation, we can assume also that $x_{n+1}=0$. Thus, we have $$C_{n+1}=x_n^2+x_1^2-{\left(x_n-x_1\right)}^2=2x_1{x}_n\ge 4.$$