7th Swedish 1967


1. p parallel lines are drawn in the plane and q lines perpendicular to them are also drawn. How many rectangles are bounded by the lines?
2. You are given a ruler with two parallel straight edges a distance d apart. It may be used (1) to draw the line through two points, (2) given two points a distance ≥ d apart, to draw two parallel lines, one through each point, (3) to draw a line parallel to a given line, a distance d away. One can also (4) choose an arbitrary point in the plane, and (5) choose an arbitrary point on a line. Show how to construct (A) the bisector of a given angle, and (B) the perpendicular to the midpoint of a given line segment.
3. Show that there are only finitely many triples (a, b, c) of positive integers such that 1/a + 1/b + 1/c = 1/1000.
4. The sequence a₁, a₂, a₃, ... of positive reals is such that ∑ a_i diverges. Show that there is a sequence b₁, b₂, b₃, ... of positive reals such that lim b_n = 0 and ∑ a_ib_i diverges.
5. a₁, a₂, a₃, ... are positive reals such that a_n² ≥ a₁ + a₂ + ... + a_n-1. Show that for some C > 0 we have a_n ≥ Cn for all n.
6. The vertices of a triangle are lattice points. There are no lattice points on the sides (apart from the vertices) and n lattice points inside the triangle. Show that its area is n + ½. Find the formula for the general case where there are also m lattice points on the sides (apart from the vertices).

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.