1997 IMO Problems/Problem 1

Problem

In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternatively 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 $f(m,n) \le \frac{1}{2} max\left\{ m,n \right\}$ for all $m$ and $n$ .

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

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

Page

Toolbox

Search

1997 IMO Problems/Problem 1

Problem

Solution