IMO Shortlist 1997 problem 1

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Dodao/la: arhiva
2. travnja 2012.
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.
Izvor: Međunarodna matematička olimpijada, shortlist 1997