SE-SKOLORNAS 1997 Problem 5

Let $s(m)$ denote the sum of (decimal) digits of a positive integer $m$ . Prove that for every integer $n > 1$ not equal to $10$ there is a unique integer $f(n) \ge 2$ such that $s(k)+s(f(n)-k) = n$ for all integers $k$ with $0 < k < f(n)$ .