13. Let $f$ be a real-valued function so that
$$
f(x, y)=f(x, z)-2 f(y, z)-2 z
$$
for all real numbers $x, y$ and $z$. Find $f(2005,1000)$.
13. Ans: 5
Setting $x=y=z$, we see that
$$
f(x, x)=f(x, x)-2 f(x, x)-2 x
$$
for all $x$. Hence $f(x, x)=-x$ for all $x$. Setting $y=x$ gives
$$
f(x, x)=f(x, z)-2 f(x, z)-2 z
$$
for all $x$ and $z$. Hence $f(x, z)=-f(x, x)-2 z=x-2 z$ for all $x$ and $z$. Therefore, $f(2005,1000)=5$.