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):

æ ç
è

x  
2 a


ö ÷
ø


æ ç
è

y  
2 a


ö ÷
ø

= a^{2}  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
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
2x^{3} + ax^{2} + 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.