Dorin Marghidanu's Sums and Products

Problem

Dorin Marghidanu's Sums and Products

Solution

We'll use Bergstrom's inequality:

$\displaystyle\begin{align}\sum_{k=1}^n\frac{a_k}{P_kS_k}&=\frac{1}{P}\sum_{k=1}\frac{a_k^2}{S_k}\\ &\ge\frac{1}{P}\cdot\frac{S^2}{nS-S}=\frac{1}{n-1}\cdot\frac{S}{P}\qquad\text{(use AM-GM)}\\ &\ge\frac{1}{n-1}\cdot\frac{S}{(S/n)^n}=\frac{n^n}{(n-1)S^{n-1}}. \end{align}$

Acknowledgment

Gabi Cucoanes has kindly messaged me his solution to a problem shared on facebook by Dorin Marghidanu. The latter was commented on with several proofs that happened to be slight modifications of the above which thus may be considered the average of the submitted proofs.

 

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