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

- 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 $u={a}^{2}+{b}^{2}+{c}^{2}$ and $v=(a+b+c){}^{2}$. Prove that

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

- 3.
- Suppose that $f(x)$ is a function satisfying

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

- 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.