Question 23
Functions $f$ and $g$ are quadratic, $g(x) = -f(100 - x),$ and the graph of $g$ contains the vertex of the graph of $f.$ The four $x$-intercepts on the two graphs have $x$-coordinates $x_1,$ $x_2,$ $x_3,$ and $x_4,$ in increasing order, and $x_3 - x_2 = 150.$ The value of $x_4 - x_1$ is $m + n\sqrt{p},$ where $m,$ $n,$ and $p$ are positive integers, and $p$ is not divisible by the square of any prime. What is $m + n + p?$