IMO 1997 Problem 1

In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured 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 $f(m,n) \leq \frac 12 \max \{m,n \}$ for all $m$ and $n$ .

c) Show that there is no constant $C\in\mathbb{R}$ such that $f(m,n) < C$ for all $m$ and $n$ .