Generalization of a RMO-2012 problem

A similar problem was given in three different regions in RMO(just the values were bit altered).I am giving it's generalization.Prove it.-

Suppose $x_{0},x_{1},..,x_{n}$ are natural numbers such that $x_{0}|x_{1}^{m}$ , $x_{1}|x_{2}^{m}$ ,..., $x_{n}|x_{0}^{m}$ where $m≥1$ is a natural number.

Then show that $∏_{i=0}^{n}x_{i}|(\sum_{i=0}^{n}x_{i})^{\frac{m^{n+1}-1}{m-1}}$ .

#NumberTheory #Inequalities #RMO

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$