10. The function $f$ has the following property: For any two real numbers $x, f(x)+f(x-1)=x^{2}$. If $f(19)=94$, then the remainder when $f(94)$ is divided by $10^{3}$ is
10. 561 .
$$
f(x+1)=(x+1)^{2}-f(x), f(x+2)=(x+2)^{2}-f(x+1)=(x+2)^{2}-(x+1)^{2}+
$$
$f(x), \cdots$ Similarly, for $m \in \mathbf{N}_{+}, f(x+m)=(x+m)^{2}-(x+m-1)^{2}+(x+m-2)^{2}+\cdots+$ $(-1)^{m-1}(x+1)^{2}+(-1)^{m} f(x)$. Therefore, $f(94)=f(19+75)=94^{2}-93^{2}+92^{2}-\cdots+20^{2}-$ $f(19)=4561$