Question 15
Let $x,$ $y,$ and $z$ be positive real numbers satisfying the system of equations $$
\sqrt{2x - xy} + \sqrt{2y - xy} = 1
$$ $$
\sqrt{2y - yz} + \sqrt{2z - yz} = \sqrt{2}
$$ $$
\sqrt{2z - zx} + \sqrt{2x - zx} = \sqrt{3}.
$$ Then $\left[(1 - x)(1 - y)(1 - z)\right]^2$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
\sqrt{2x - xy} + \sqrt{2y - xy} = 1
$$ $$
\sqrt{2y - yz} + \sqrt{2z - yz} = \sqrt{2}
$$ $$
\sqrt{2z - zx} + \sqrt{2x - zx} = \sqrt{3}.
$$ Then $\left[(1 - x)(1 - y)(1 - z)\right]^2$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$