location:
PROBLEMS FOR DECEMBER

Solutions should be submitted to

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

no later than January 31, 2001.

49.
Find all ordered pairs (x, y) that are solutions of the following system of two equations (where a is a parameter):
 x - y = 2
 æç è x - 2a ö÷ ø æç è y - 2a ö÷ ø = a2 - 1 .
Find all values of the parameter a for which the solutions of the system are two pairs of nonnegative numbers. Find the minimum value of x + y for these values of a.

50.
Let n be a natural number exceeding 1, and let An be the set of all natural numbers that are not relatively prime with n (i.e., An = { x Î N :  gcd (x, n) ¹ 1 }. Let us call the number n magic if for each two numbers x, y Î An, their sum x + y is also an element of An (i.e., x + y Î An for x, y Î An).
(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.
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