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.


S= 12 1·3 + 22 3·5 + 32 5·7 ++ 5002 999·1001 .

Find the value of S.

The sequences { an } and { bn } are such that, for every positive integer n,

an >0, bn >0, an+1 = an + 1 bn , bn+1 = bn + 1 an .

Prove that a50 + b50 >20.

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.

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.

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.

Each vertex of a regular 100-sided polygon is marked with a number chosen from among the natural numbers 1,2,3,,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.