MathDB

Let

x

y

, and

z

be positive real numbers satisfying the system of equations \begin{align*} \sqrt{2x - xy} + \sqrt{2y - xy} & = 1\\ \sqrt{2y - yz} + \hspace{0.1em} \sqrt{2z - yz} & = \sqrt{2}\\ \sqrt{2z - zx\vphantom{y}} + \sqrt{2x - zx\vphantom{y}} & = \sqrt{3}. \end{align*}Then

\big[ (1-x)(1-y)(1-z) \big] ^2

can be written as

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m+n

Problem Statement