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):
$xy=2$
$(x\frac{2}{a})(y\frac{2}{a})={a}^{2}1\hspace{1em}.$
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.,
${A}_{n}=\{x\in N:\hspace{1em}\mathrm{gcd}\hspace{1em}(x,n)\ne 1\}$.
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
$x,y\in {A}_{n}$).


(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
$\mathrm{ABC}$,
$\mathrm{AB}=15$,
$\mathrm{BC}=13$ and
$\mathrm{AC}=12$. 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
$2{x}^{3}+{\mathrm{ax}}^{2}+\mathrm{bx}+8=0$
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
$(2n+1)$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.