Solutions should be submitted to

Dr. Valeria Pandelieva
708 - 195 Clearview Avenue
Ottawa, ON K1Z 6S1

49.

Find all ordered pairs (x, y) that are solutions of the following system of two equations (where a is a parameter):

50.

Let n be a natural number exceeding 1, and let A_n be the set of all natural numbers that are not relatively prime with n (i.e., A_n = { x Î N :  gcd (x, n) ¹ 1 }. Let us call the number n magic if for each two numbers x, y Î A_n, their sum x + y is also an element of A_n (i.e., x + y Î A_n for x, y Î A_n).

(a) Prove that 67 is a magic number.

(b) Prove that 2001 is not a magic number.

(c) Find all magic numbers.

51.

In the triangle ABC, AB = 15, BC = 13 and AC = 12. Prove that, for this triangle, the angle bisector from A, the median from B and the altitude from C are concurrent (i.e., meet in a common point).

52.

One solution of the equation 2x³ + ax² + bx + 8 = 0 is 1 + Ö3. Given that a and b are rational numbers, determine its other two solutions.

53.

Prove that among any 17 natural numbers chosen from the sets { 1, 2, 3, ¼, 24, 25 }, it is always possible to find two whose product is a perfect square.

54.

A circle has exactly one common point with each of the sides of a (2n+1)-sided polygon. None of the vertices of the polygon is a point of the circle. Prove that at least one of the sides is a tangent of the circle.