MathDB

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}.

Problem Statement