100,000 points problem

$\frac{3}{x-3}+1+\frac{5}{x-5}+1+\frac{17}{x-17}+1+\frac{19}{x-19}+1=x^2-11x$ $x(\color{#3D99F6}{\frac{1}{x-3}}+\color{#D61F06}{\frac{1}{x-5}+\frac{1}{x-17}}\color{#3D99F6}{+\frac{1}{x-19}})=x(x-11)$ $2(x-11)(\frac{1}{(x-3)(x-19)}+\frac{1}{(x-5)(x-17)})=x-11$ $\color{#20A900}{\text{Let}\space x^{2}-22x=t \Rightarrow x=11\pm\sqrt{t+121}}$ $\Rightarrow \frac{2}{t+57}+\frac{2}{t+85}=1$ $\Rightarrow t^2+138t+6885=0 \Rightarrow t=-69\pm \sqrt{200}$ $\color{#69047E}{m=11+ \sqrt{ (-69+\sqrt{200})+121} }$ $=11+\sqrt { (52+\sqrt{200} })$ $a+b+c=11+52+200=\color{#3D99F6}{263}$

first off.. i loved this problem, thanks to the creator for giving me such great experience. start like this $\begin{aligned} \frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}+4= x^2-11x\\ \frac{3}{x-3}+1+\frac{5}{x-5}+1+\frac{17}{x-17}+1+\frac{19}{x-19}+1= x^2-11x\\ \frac{x}{x-3}+\frac{x}{x-5}+\frac{x}{x-17}+\frac{x}{x-19}=x(x-11)\end{aligned}$ we get x=0 as a solution. now divide by x and put x=y+11 $\begin{aligned} \frac{1}{y+8}+\frac{1}{y-8}+\frac{1}{y+6}+\frac{1}{y-6}=y\\ \frac{2y}{y^2-64}+\frac{2y}{y^2-36}=y \end{aligned}$ y=0 ->x=11 is another solution. we divide again to get $\begin{aligned} \frac{2}{y^2-64}+\frac{2}{y^2-36}=1\\ 2(y^2-36)+2(y^2-64)=(y^2-64)(y^2-36)\\ 4y^2-200=y^4-100y^2+2304\\ y^4-104y^2+2504=0\\ y^2=52\pm \sqrt{200}\\ y=\pm\sqrt{52\pm \sqrt{200}}\\ x=11\pm\sqrt{52\pm \sqrt{200}}\end{aligned}$ we get that $x=0,11,11\pm\sqrt{52\pm \sqrt{200}}$ the required $x_{largest}=11+\sqrt{52+ \sqrt{200}}$ $11+52+200=\boxed{263}$

I just want to say that it would have been fun if it was similar to the original problem but not exactly the same.