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.
Find all ordered pairs
that are solutions
of the following system of two equations (where
Find all values of the parameter
for which the solutions of
the system are two pairs of nonnegative numbers. Find the
minimum value of
for these values of
be a natural number exceeding 1, and let
be the set of all natural numbers that are
not relatively prime with
Let us call the number
magic if for each two numbers
, their sum
is also an element of
(a) Prove that 67 is a magic number.
(b) Prove that 2001 is not a magic number.
(c) Find all magic numbers.
In the triangle
. Prove that, for this triangle, the angle bisector
, the median from
and the altitude from
concurrent (i.e., meet in a common point).
One solution of the equation
. Given that
numbers, determine its other two solutions.
Prove that among any 17 natural numbers chosen from
, it is always possible
to find two whose product is a perfect square.
A circle has exactly one common point with each of the
sides of a
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.