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**.

- 7.
- Let

Find the value of $S$.

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

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 $a+b=c+d$.