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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.

Slični zadaci

#NaslovOznakeRj.KvalitetaTežina
1982IMO Shortlist 1997 problem 260
1975IMO Shortlist 1997 problem 192
1969IMO Shortlist 1997 problem 132
1967IMO Shortlist 1997 problem 111
1246IMO Shortlist 1966 problem 630
1242IMO Shortlist 1966 problem 590