VN-TST 1991 Problem 2

For a positive integer $n>2$ , let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a sequence of $n$ positive reals which is either non-decreasing (this means, we have $a_{1}\leq a_{2}\leq \ldots \leq a_{n}$ ) or non-increasing (this means, we have $a_{1}\geq a_{2}\geq \ldots \geq a_{n}$ ), and which satisfies $a_{1}\neq a_{n}$ . Let $x$ and $y$ be positive reals satisfying $\frac{x}{y}\geq \frac{a_{1}-a_{2}}{a_{1}-a_{n}}$ . Show that: $$\frac{a_{1}}{a_{2}\cdot x+a_{3}\cdot y}+\frac{a_{2}}{a_{3}\cdot x+a_{4}\cdot y}+\ldots+\frac{a_{n-1}}{a_{n}\cdot x+a_{1}\cdot y}+\frac{a_{n}}{a_{1}\cdot x+a_{2}\cdot y}\geq \frac{n}{x+y}.$$