Let's Make 2018 Special!

We have $f(x) = \frac3{x-22} + \frac7{x-57} + \cdots + \frac{a_{2018}}{x-b_{2018}}$ where $a_{2018} = 2018^2+2018+1$ and $b_{2018} = 3(2018)^3+3(2018)^2+5(2018)+11.$ Note that $3k^3+3k^2+5k+11$ is an increasing function for $k>0,$ so each of the integers in the denominator is increasing.

Note also that $f(x)$ is strictly decreasing on its domain (take a derivative; each of the terms in the sum is negative).

So there are $2018$ vertical asymptotes to $y=f(x).$ For $x<22,$ $f(x)$ is negative. For $22<x<57,$ $f(x)$ decreases from $+\infty$ to $-\infty.$ So it crosses the line $y=99$ exactly once. Ditto for $57 < x < 134.$ And so on. Finally, for $x > b_{2018},$ $y=f(x)$ decreases from $+\infty$ to approaching $0$ as $x\to\infty,$ so it crosses the line $y=99$ exactly once there as well.

So there are $2017$ roots, one between each consecutive pair of vertical asymptotes; and one more root to the right of the righmost vertical asymptote. This is a total of $\fbox{2018}.$