PROBLEMS FOR OCTOBER

Please send your solutions to

Valeria Pandelieva
641 Kirkwood Avenue
Ottawa, ON K1Z 5X5

109.

Suppose that

x2 + y2 x2 - y2 + x2 - y2 x2 + y2 = k .

Find, in terms of k, the value of the expression

x8 + y8 x8 - y8 + x8 - y8 x8 + y8  .

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 ÇCM, {R} = BQ ÇAN and {S} = CM ÇBQ, where Ç 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 a. A line passing through the vertex A is perpendicular to the angle bisector of ÐBAC and intersects the line BC at the point M. Find the other two angles of the triangle ABC in terms of a, if it is known that BM = BA + AC.

113.

Find a function that satisfies all of the following conditions:

(a) f is defined for every positive integer n;

(b) f takes only positive values;

(c) f(4) = 4;

(d)

1 f(1)f(2) + 1 f(2)f(3) + ¼+ 1 f(n)f(n+1) = 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.