10. Let $f(x, y)$ be a quadratic polynomial, and (1) $f(1,2)=2$; (2) $y f(x, f(x, y))=x \cdot f(f(x, y), y)=(f(x, y))^{2}$. Find all $f(x, y)$.
10. $f(x, y)=x y$ Let the highest power of $x$ in $f(x, y)$ be $m$, then the highest power of $x$ in $(f(x, y))^{2}$ is $2 m$. Also, the highest power of $x$ in $x \cdot f(f(x, y), y)$ is $1+m^{2}$, hence $2 m=1+m^{2}, m=1$. Similarly, the highest power of $y$ in $f(x, y)$ is 1. We can assume $f(x, y)=a x y+b x+c y+d$. Since $x, y\left|f(x, y)^{2}, x, y\right| f(x, y)$, then $b=c=$ $d=0 . f(x, y)=a x y$, from (1) we know, $a=1, f(x, y)=x y$.