HMMT February, 2019

Solution:
Letting $y=-x$ gives
$$f\left(x^{2}\right)+f\left(x^{2}\right)=f(0)^{2}+2 x^{2}$$
for all $x$. When $x=0$ the equation above gives $f(0)=0$ or $f(0)=2$.
If $f(0)=2$, then $f(x)=x+2$ for all nonegative $x$, so the LHS becomes $x^{2}+y^{2}+4$, and RHS becomes $x^{2}+y^{2}+4 x+4 y+4$ for all $x+y \geq 0$, which cannot be equal to LHS if $x+y>0$.
If $f(0)=0$ then $f(x)=x$ for all nonnegative $x$. Moreover, letting $y=0$ gives
$$f\left(x^{2}\right)=f(x)^{2} \Rightarrow f(x)= \pm x$$
for all $x$. Since negative values are never used as inputs on the LHS and the output on the RHS is always squared, we may conclude that for all negative $x, f(x)=x$ and $f(x)=-x$ are both possible (and the values are independent). Therefore, the value of $S$ can be written as
$$S=f(0)+(f(1)+f(-1))+(f(2)+f(-2))+\cdots+(f(2019)+f(-2019))=2 \sum_{i=1}^{2019} i \delta_{i}$$
for $\delta_{1}, \delta_{2}, \ldots, \delta_{2019} \in\{0,1\}$. It is not difficult to see that $\frac{S}{2}$ can take any integer value between 0 and $\frac{2020 \cdot 2019}{2}=2039190$ inclusive, so there are 2039191 possible values of $S$.