PROBLEMS FOR JULY
Please send your solutions to
Dr. Valeria Pandelieva
641 Kirkwood Avenue
Ottawa, ON K1Z 5X5
no later than August 31, 2001 and no sooner than
August 15, 2001.
Note. There was an unfortunate error in the statement
of Problem 77. I would like to apologize to students
who tried to solve the problem and did not get the point of
it because of the mistake. A corrected version is listed below, and
solutions can be mailed to Dr. Pandelieva. Some of the
original solvers detected the error and sent solutions to
the problem that was intended; such students need not send
anything further on this problem. (If the statement of
a problem on a competition seems fishy, draw attention
to what you think may be the probably error, explicitly
state a nontrivial formulation of the problem and
solve that.) (E. Barbeau)

77.

$n$ points are chosen from the circumference
or the interior of a regular hexagon with sides of unit
length, so that the distance between any two of them is
not less that
$\sqrt{2}$. What is the largest
natural number
$n$ for which this is possible?

91.

A square and a regular pentagon are inscribed in
a circle. The nine vertices are all distinct and divide the
circumference into nine arcs. Prove that at least one of them
does not exceed 1/40 of the circumference of the circle.

92.

Consider the sequence
$200125$,
$2000125$,
$20000125$,
$\dots $,
$200\dots 00125$,
$\dots $ (in which the
$n$th number has
$n+1$ digits equal to zero).
Can any of these numbers be the square or the cube of
an integer?

93.

For any natural number
$n$, prove the
following inequalities:
${2}^{(n1)/({2}^{n2})}\le \sqrt{2}\sqrt[4]{4}\sqrt[8]{8}\dots \sqrt[{2}^{n}]{{2}^{n}}<4\hspace{1em}.$

94.

$\mathrm{ABC}$ is a right triangle with arms
$a$ and
$b$ and hypotenuse
$c=\Vert \mathrm{AB}\Vert $; the area of the
triangle is
$s$ square units and its perimeter is
$2p$ units. The numbers
$a$,
$b$ and
$c$ are positive integers.
Prove that
$s$ and
$p$ are also positive integers and that
$s$ is a multiple of
$p$.

95.

The triangle
$\mathrm{ABC}$ is isosceles with
equal sides
$\mathrm{AC}$ and
$\mathrm{BC}$. Two of its angles measure
${40}^{\u02c6}$. The interior point
$M$ is such that
$\angle \mathrm{MAB}={10}^{\u02c6}$ and
$\angle \mathrm{MBA}={20}^{\u02c6}$.
Determine the measure of
$\angle \mathrm{CMB}$.

96.

Find all prime numbers
$p$ for which all three
of the numbers
${p}^{2}2$,
$2{p}^{2}1$ and
$3{p}^{2}+4$ are also
prime.