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# 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 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 $u={a}^{2}+{b}^{2}+{c}^{2}$ and $v=\left(a+b+c\right){}^{2}$. 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 $f\left(x\right)$ is a function satisfying

$‖f\left(m+n\right)-f\left(m\right)‖\le \frac{n}{m}$

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

$\sum _{i=1}^{k}‖f\left({2}^{k}\right)-f\left({2}^{i}\right)‖\le \frac{k\left(k-1\right)}{2} .$

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.
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