PROBLEMS FOR OCTOBER

Canadian Mathematical Society

www.cms.math.ca

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

location:

Please send your solutions to

Valeria Pandelieva

641 Kirkwood Avenue

Ottawa, ON K1Z 5X5

\frac{x^{2} + y^{2}}{x^{2} - y^{2}} + \frac{x^{2} - y^{2}}{x^{2} + y^{2}} = k .

Find, in terms of

k

, the value of the expression

\frac{x^{8} + y^{8}}{x^{8} - y^{8}} + \frac{x^{8} - y^{8}}{x^{8} + y^{8}} .

110.: Given a triangle $ABC$ with an area of 1. Let $n > 1$ be a natural number. Suppose that $M$ is a point on the side $AB$ with $AB = nAM$ , $N$ is a point on the side $BC$ with $BC = nBN$ , and $Q$ is a point on the side $CA$ with $CA = nCQ$ . Suppose also that ${T} = AN \cap CM$ , ${R} = BQ \cap AN$ and ${S} = CM \cap BQ$ , where $\cap$ signifies that the singleton is the intersection of the indicated segments. Find the area of the triangle $TRS$ in terms of $n$ .

111.: (a) Are there four different numbers, not exceeding 10, for which the sum of any three is a prime number?

: (b) Are there five different natural numbers such that the sum of every three of them is a prime number?

112.: Suppose that the measure of angle $BAC$ in the triangle $ABC$ is equal to $α$ . A line passing through the vertex $A$ is perpendicular to the angle bisector of $∠ BAC$ and intersects the line $BC$ at the point $M$ . Find the other two angles of the triangle $ABC$ in terms of $α$ , if it is known that $BM = BA + AC$ .

\frac{1}{f (1) f (2)} + \frac{1}{f (2) f (3)} + \dots + \frac{1}{f (n) f (n + 1)} = \frac{f (n)}{f (n + 1)} .

114.: A natural number is a multiple of 17. Its binary representation (i.e., when written to base 2) contains exactly three digits equal to 1 and some zeros.

: (a) Prove that there are at least six digits equal to 0 in its binary representation.

: (b) Prove that, if there are exactly seven digits equal to 0 and three digits equal to 1, then the number must be even.