CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
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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- 2 a ) (y- 2 a )= 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 ={xN:gcd(x,n)1}. Let us call the number n magic if for each two numbers x,y An , their sum x+y is also an element of An (i.e., x+y An for x,y 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 2 x3 + ax2 +bx+8=0 is 1+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,,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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