jmc

Subtracting $10000\lfloor x\rfloor$ from both sides, we get the equation $$x^2=10000(x-\lfloor x\rfloor )=10000\{x\}.$$To understand the solutions to this equation, we consider the graphs of $y=x^2$ and $y=10000\{x\}.$ The graph of $y=x^2$ is the usual parabola; the graph of $y=10000\{x\}$ consists of line segments between the points $(n,0)$ and $(n+1,10000)$ for each integer $n,$ including the left endpoint but not the right endpoint: Note that the graph of $y=x^2$ passes through both the points $P=(-100,10000)$ and $Q=(100,10000),$ as shown above, so all of the points $(-99,10000),$ $(-98,10000),$ $\dots ,$ $(99,10000)$ lie above the parabola. It follows that the parabola intersects only the segments corresponding to those points. There are $99-(-99)+1=199$ of these segments, so the number of solutions to the equation is $\boxed{199}.$