PROBLEMS FOR APRIL
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 wordprocessing 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
$u={a}^{2}+{b}^{2}+{c}^{2}$ and
$v=(a+b+c){}^{2}$.
Prove that
$\frac{1}{3}\le \frac{u}{v}<\frac{1}{2}$
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
$\Vert f(m+n)f(m)\Vert \le \frac{n}{m}$
for all rational numbers
$n$ and
$m$. Show that, for all
natural numbers
$k$,
$\sum _{i=1}^{k}\Vert f({2}^{k})f({2}^{i})\Vert \le \frac{k(k1)}{2}\hspace{1em}.$

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.