ISL 1991 Problem 23

Let $f$ and $g$ be two integer-valued functions defined on the set of all integers such that

(a) $f(m + f(f(n))) = -f(f(m+ 1) - n$ for all integers $m$ and $n;$ (b) $g$ is a polynomial function with integer coefficients and g(n) = $g(f(n))$ $\forall n \in \mathbb{Z}.$