Canadian Mathematical Society

www.cms.math.ca

	\|		\|	Français
	\|	Site map	\|	CMS store

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.

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

Find the value of

S

8.: The sequences ${a_{n}}$ and ${b_{n}}$ 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 $ABCD$ is a rectangle, the points $A$ and $B$ are two adjacent vertices of the rectangle and $a + b = c + d$ .