jmc

Let $$f(x)=\frac{1}{x-1}+\frac{2}{x-2}+\frac{3}{x-3}+\cdots +\frac{100}{x-100}.$$Consider the graph of $y=f(x).$



The graph of $y=f(x)$ has vertical asymptotes at $x=1,$ $x=2,$ $\dots ,$ $x=100.$ In particular, $f(x)$ approaches $-\infty$ as $x$ approaches $n$ from the left, and $f(x)$ approaches $\infty$ as $x$ approaches $n$ from the right, for $1\le n\le 100.$ Furthermore, $y=0$ is a vertical asymptote. In particular, $f(x)$ approaches 0 as $x$ approaches both $\infty$ and $-\infty .$

Thus, the graph of $y=f(x)$ intersects the graph of $y=x$ exactly once on each of the intervals $(-\infty ,1),$ $(1,2),$ $(2,3),$ $\dots ,$ $(99,100),$ $(100,\infty ).$ Therefore, there are a total of $\boxed{101}$ real solutions.