7th Swedish Mathematical Society Problems 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). 

Solutions

Problem 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?

Answer

pq(p-1)(q-1)/4

Solution

The lines give a p x q array of possible vertices. We pick any vertex, then any vertex not in the same row or column. That defines a rectangle. It can be done in pq(p-1)(q-1) ways and gives each rectangle 4 times. 

Problem 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.

Solution

wlog a ≤ b ≤ c. Then a ≤ 3000, so there are only finitely many possible values for a. Now for given a we have 1/b + 1/c = fixed. Again there are only a finite number of possibilities for b, and c is then fixed. 

Problem 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.

Solution

Put b_n = 1/(a₁ + a₂ + ... + a_n). Then lim b_n = 0. Given any m we can find n such that s_n > 2s_m and hence s_k > 2s_m for any k > m. Hence a_m+1b_m+1 + ... + a_kb_k ≥ (s_k - s_m)/s_k = 1 - s_m/s_k > 1/2 for any k > n. Hence ∑ a_nb_n diverges.

Thanks to Thomas Linhart

Problem 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).

Answer

The area of any polygon with all vertices at lattice points (and boundary not self-intersecting) is ½b + c - 1, where b is the number of lattice points on the boundary (including vertices) and c is the number of lattice points inside.

Solution

This is just Pick's theorem, which is bookwork.