APMO 2002 Problem 5

Let R denote the set of all real numbers. Find all functions $f$ from R to R satisfying: (i) there are only finitely many $s$ in R such that $f(s)=0$, and (ii) $f\left(x^{4}+y\right)=x^{3} f(x)+f(f(y))$ for all $x, y$ in $\mathbf{R}$.