location:

# 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 $\left(x,y\right)$ that are solutions of the following system of two equations (where $a$ is a parameter):

$x-y=2$

$\left(x-\frac{2}{a}\right)\left(y-\frac{2}{a}\right)={a}^{2}-1 .$

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., ${A}_{n}=\left\{x\in N: \mathrm{gcd} \left(x,n\right)\ne 1\right\}$. 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 $\left\{1,2,3,\dots ,24,25\right\}$, 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 $\left(2n+1\right)-$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.