PROBLEMS FOR FEBRUARY
Please send your solutions to
Valeria Pandelieva
641 Kirkwood Avenue
Ottawa, ON K1Z 5X5
no later than March 15, 2002.

127.

Let
$A={2}^{n}+{3}^{n}+216({2}^{n6}+{3}^{n6})$
and
$B={4}^{n}+{5}^{n}+8000({4}^{n6}+{5}^{n6})$
where
$n>6$ is a natural number. Prove that the
fraction
$A/B$ is reducible.

128.

Let
$n$ be a positive integer. On a circle,
$n$ points are marked. The number 1 is assigned to one
of them and 0 is assigned to the others. The following
operation is allowed: Choose a point to which 1 is assigned
and then assign
$(1a)$ and
$(1b)$ to the two adjacent
points, where
$a$ and
$b$ are, respectively, the numbers
assigned to these points before. Is it possible to assign
1 to all points by applying this operation several times
if (a)
$n=2001$ and (b)
$n=2002$?

129.

For every integer
$n$, a nonnegative integer
$f(n)$ is assigned such that


(a)
$f(\mathrm{mn})=f(m)+f(n)$ for each pair
$m,n$
of natural numbers;


(b)
$f(n)=0$ when the rightmost digit in the
decimal representation of the number
$n$ is 3; and


(c)
$f(10)=0$.


Prove that
$f(n)=0$ for any natural number
$n$.

130.

Let
$\mathrm{ABCD}$ be a rectangle for which the
respective lengths of
$\mathrm{AB}$ and
$\mathrm{BC}$ are
$a$ and
$b$.
Another rectangle is circumscribed around
$\mathrm{ABCD}$ so that
each of its sides passes through one of the vertices of
$\mathrm{ABCD}$. Consider all such rectangles and, among them,
find the one with a maximum area. Express this area in
terms of
$a$ and
$b$.

131.

At a recent winter meeting of the Canadian
Mathematical Society, some of the attending mathematicians
were friends. It appeared that every two mathematicians, that
had the same number of friends among the participants, did
not have a common friend. Prove that there was a mathematician
who had only one friend.

132.

Simplify the expression
$\sqrt[5]{3\sqrt{2}2\sqrt{5}}\xb7\sqrt[10]{\frac{6\sqrt{10}+19}{2}}\hspace{1em}.$