1

Part of 1992 Romania Team Selection Test

Problems(2)

increasing f : N \to N , t f(f(n)) = 3n , f(1992) =?

Source: Romania BMO TST 1992 p1

f : N \to N

is an increasing function such that

f(f(n)) = 3n

n

f(1992)

functionalfunctionalgebra

there exist k vertices of these rectangles which lie on a line

Source: Romania IMO TST 1992 p1

S > 1

be a real number. The Cartesian plane is partitioned into rectangles whose sides are parallel to the axes of the coordinate system. and whose vertices have integer coordinates. Prove that if the area of each triangle if at most

S

, then for any positive integer

k

k

vertices of these rectangles which lie on a line.

geometryrectanglecombinatoricscollinearcombinatorial geometry