Have you tried Rearrangement Inequality?

Note: the problem in question is the following:

Let $a_i>0 (i=1,2,\ldots,n)$, and real numbers $x,y$ satisfy $xy>0$. Show
$$
\frac{a_1^x}{a_2^y}+\frac{a_2^x}{a_3^y}+\cdots+\frac{a_{n-1}^x}{a_n^y}+\frac{a_n^x}{a_1^y} \geq \sum_{i=1}^n a_i^{x-y}.
$$

(Edited by Areteem Institute based on comment from William Chen- original submission Friday, June 15, 2018, 11:56 PM)