jmc

Setting $x=10$ and $y=5$ gives $f(10)+f(25)+250=f(25)+200+1$, from which we get $f(10)=\boxed{-49}$.

$\text{Remark:}$ By setting $y=\frac{x}{2}$, we see that the function is $f(x)=-\frac{1}{2}{x}^2+1$, and it can be checked that this function indeed satisfies the given equation.