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 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.

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.

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

1 3 u v < 1 2

and that the fraction 1/2 on the right cannot be replaced by a smaller number.

Suppose that f(x) is a function satisfying

f(m+n)-f(m) n m

for all rational numbers n and m. Show that, for all natural numbers k,

i=1 kf( 2k )-f( 2i ) k(k-1) 2 .

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

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.

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.