Question 20
Let $x$, $y$, and $z$ be positive real numbers satisfying the system of equations
$$
\begin{aligned}
\sqrt{2x - xy} + \sqrt{2y - xy} &= 1 \\
\sqrt{2y - yz} + \sqrt{2z - yz} &= \sqrt{2} \\
\sqrt{2z - zx} + \sqrt{2x - zx} &= \sqrt{3}.
\end{aligned}
$$
Then $[(1-x)(1-y)(1-z)]^2$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
$$
\begin{aligned}
\sqrt{2x - xy} + \sqrt{2y - xy} &= 1 \\
\sqrt{2y - yz} + \sqrt{2z - yz} &= \sqrt{2} \\
\sqrt{2z - zx} + \sqrt{2x - zx} &= \sqrt{3}.
\end{aligned}
$$
Then $[(1-x)(1-y)(1-z)]^2$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.