CWMO 2018 Problem 6

Let $n \geq 2$ be an integer. Positive reals satisfy $a_1\geq a_2\geq \cdots\geq a_n.$ Prove that $$\left(\sum_{i=1}^n\frac{a_i}{a_{i+1}}\right)-n \leq \frac{1}{2a_1a_n}\sum_{i=1}^n(a_i-a_{i+1})^2,$$ where $a_{n+1}=a_1.$