PROBLEMS FOR MAY
Solutions should be submitted to
Dr. Valeria Pandelieva
708  195 Clearview Avenue
Ottawa, ON K1Z 6S1
Solution to these problems should be
postmarked no later than June 30, 2000.
Notes: A set of lines of concurrent
if and only if they have a common point of intersection.

7.

Let
$S=\frac{{1}^{2}}{1\xb73}+\frac{{2}^{2}}{3\xb75}+\frac{{3}^{2}}{5\xb77}+\dots +\frac{{500}^{2}}{999\xb71001}\hspace{1em}.$
Find the value of
$S$.

8.

The sequences
$\{{a}_{n}\}$ and
$\{{b}_{n}\}$ are such
that, for every positive integer
$n$,
${a}_{n}>0\hspace{1em},\hspace{1em}\hspace{1em}\hspace{1em}{b}_{n}>0\hspace{1em},\hspace{1em}\hspace{1em}\hspace{1em}{a}_{n+1}={a}_{n}+\frac{1}{{b}_{n}}\hspace{1em},\hspace{1em}\hspace{1em}\hspace{1em}{b}_{n+1}={b}_{n}+\frac{1}{{a}_{n}}\hspace{1em}.$
Prove that
${a}_{50}+{b}_{50}>20$.

9.

There are six points in the plane. Any three of them
are vertices of a triangle whose sides are of different length.
Prove that there exists a triangle whose smallest side is the
largest side of another triangle.

10.

In a rectangle, whose sides are 20 and 25 units of
length, are placed 120 squares of side 1 unit of length.
Prove that a circle of diameter 1 unit can be placed in the
rectangle, so that it has no common points with the squares.

11.

Each of nine lines divides a square into two quadrilaterals,
such that the ratio of their area is 2:3. Prove that at
least three of these lines are concurrent.

12.

Each vertex of a regular 100sided polygon is
marked with a number chosen from among the natural numbers
$1,2,3,\dots ,49$. Prove that there are four
vertices (which we can denote as
$A$,
$B$,
$C$,
$D$ with
respective numbers
$a$,
$b$,
$c$,
$d$) such that
$\mathrm{ABCD}$ is a rectangle, the points
$A$ and
$B$ are two
adjacent vertices of the rectangle and
$a+b=c+d$.