Solutions should be submitted to Prof. E.J. Barbeau, Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3. Electronic solutions should be submitted either in plain text or in TeX; please do not use other word-processing software; the email address is barbeau@math.utoronto.ca. Solutions should be delivered by hand or mail with a postmark no later than May 31, 2000.

Notes: The inradius of a triangle is the radius of the incircle, the circle that touches each side of the polygon. The circumradius of a triangle is the radius of the circumcircle, the circle that passes through its three vertices.

1.

Let M be a set of eleven points consisting of the four vertices along with seven interior points of a square of unit area.

(a) Prove that there are three of these points that are vertices of a triangle whose area is at most 1/16.

(b) Give an example of a set M for which no four of the interior points are collinear and each nondegenerate triangle formed by three of them has area at least 1/16.

2.

Let a, b, c be the lengths of the sides of a triangle. Suppose that u = a² + b² + c² and v = (a + b + c)². Prove that

3.

Suppose that f(x) is a function satisfying

4.

Is it true that any pair of triangles sharing a common angle, inradius and circumradius must be congruent?

5.

Each point of the plane is coloured with one of 2000 different colours. Prove that there exists a rectangle all of whose vertices have the same colour.

6.

Let n be a positive integer, P be a set of n primes and M a set of at least n+1 natural numbers, each of which is divisible by no primes other than those belonging to P. Prove that there is a nonvoid subset of M, the product of whose elements is a square integer.