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

8.

The sequences { a_n } and { b_n } are such that, for every positive integer n,

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, ¼, 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 ABCD is a rectangle, the points A and B are two adjacent vertices of the rectangle and a + b = c + d.