Question 15
Given that $x,$ $y,$ and $z$ are real numbers that satisfy $$
x = \sqrt{y^2 - \frac{1}{16}} + \sqrt{z^2 - \frac{1}{16}},
$$ $$
y = \sqrt{z^2 - \frac{1}{25}} + \sqrt{x^2 - \frac{1}{25}},
$$ $$
z = \sqrt{x^2 - \frac{1}{36}} + \sqrt{y^2 - \frac{1}{36}},
$$ and that $x + y + z = \frac{m}{\sqrt{n}},$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime, find $m + n.$
x = \sqrt{y^2 - \frac{1}{16}} + \sqrt{z^2 - \frac{1}{16}},
$$ $$
y = \sqrt{z^2 - \frac{1}{25}} + \sqrt{x^2 - \frac{1}{25}},
$$ $$
z = \sqrt{x^2 - \frac{1}{36}} + \sqrt{y^2 - \frac{1}{36}},
$$ and that $x + y + z = \frac{m}{\sqrt{n}},$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime, find $m + n.$