Question 14
Let $m$ be the largest real solution to the equation $$
\begin{aligned} &\frac{3}{x-3} + \frac{5}{x-5} \\ &\quad {}+ \frac{17}{x-17} + \frac{19}{x-19} \\ &= x^2 - 11x - 4. \end{aligned}
$$ There are positive integers $a,$ $b,$ and $c$ such that $m = a + \sqrt{b + \sqrt{c}}.$ Find $a + b + c.$
\begin{aligned} &\frac{3}{x-3} + \frac{5}{x-5} \\ &\quad {}+ \frac{17}{x-17} + \frac{19}{x-19} \\ &= x^2 - 11x - 4. \end{aligned}
$$ There are positive integers $a,$ $b,$ and $c$ such that $m = a + \sqrt{b + \sqrt{c}}.$ Find $a + b + c.$