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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

(x - \frac{2}{a}) (y - \frac{2}{a}) = 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., $A_{n} = {x \in N : \gcd (x, n) \neq 1}$ . Let us call the number $n$ magic if for each two numbers $x, y \in A_{n}$ , their sum $x + y$ is also an element of $A_{n}$ (i.e., $x + y \in A_{n}$ for $x, y \in A_{n}$ ).

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 $2 x^{3} + {ax}^{2} + bx + 8 = 0$ is $1 + \sqrt{3}$ . 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 ${1, 2, 3, \dots, 24, 25}$ , 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 $(2 n + 1) -$ 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.