smc

If $x=x^2+y^2$ and $y=2xy$, then we can break this into two cases. Case 1: $y=0$ If $y=0$, then $x=x^2$ and $0=0$ Therefore, $x=0$ or $x=1$ This yields 2 solutions Case 2: $x=\frac{1}{2}$ If $x=\frac{1}{2}$, this means that $y=y$, and $\frac{1}{2}=\frac{1}{4}+y^2$. Because y can be negative or positive, this yields $y=\frac{1}{2}$ or $y=-\frac{1}{2}$ This yields another 2 solutions. $2+2=\boxed{4}$