International Mathematical Olympiad 1997 Problem 1

In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternately black and white (as on a chessboard).

For any pair of positive integers $m$ and $n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $m$ and $n$, lie along edges of the squares.

Let $S_1$ be the total area of the black part of the triangle and $S_2$ be the total area of the white part. Let

$$f(m,n) = |S_1 - S_2|.$$

(a) Calculate $f(m,n)$ for all positive integers $m$ and $n$ which are either both even or both odd.

(b) Prove that $f(m,n) \leq \frac{1}{2}\max\{m,n\}$ for all $m$ and $n$.

(c) Show that there is no constant $C$ such that $f(m,n) < C$ for all $m$ and $n$.