location:
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 = 121 ·3 + 223 ·5 + 325 ·7 + ¼+ 5002999·1001 .
Find the value of S.

8.
The sequences { an } and { bn } are such that, for every positive integer n,
 an > 0 ,   bn > 0 ,   an+1 = an + 1bn ,   bn+1 = bn + 1an .
Prove that a50 + b50 > 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, 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.
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