PROBLEMS FOR MAY

Canadian Mathematical Society
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# 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

$S=\frac{{1}^{2}}{1·3}+\frac{{2}^{2}}{3·5}+\frac{{3}^{2}}{5·7}+\dots +\frac{{500}^{2}}{999·1001} .$

Find the value of $S$.

8.
The sequences $\left\{{a}_{n}\right\}$ and $\left\{{b}_{n}\right\}$ are such that, for every positive integer $n$,

${a}_{n}>0 , {b}_{n}>0 , {a}_{n+1}={a}_{n}+\frac{1}{{b}_{n}} , {b}_{n+1}={b}_{n}+\frac{1}{{a}_{n}} .$

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 100-sided 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 $a+b=c+d$.
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