location:

# PROBLEMS FOR JULY

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:

${2}^{\left(n-1\right)/\left({2}^{n-2}\right)}\le \sqrt{2}\sqrt[4]{4}\sqrt[8]{8}\dots \sqrt[{2}^{n}]{{2}^{n}}<4 .$

94.
$\mathrm{ABC}$ is a right triangle with arms $a$ and $b$ and hypotenuse $c=‖\mathrm{AB}‖$; 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 ${40}^{ˆ}$. The interior point $M$ is such that $\angle \mathrm{MAB}={10}^{ˆ}$ and $\angle \mathrm{MBA}={20}^{ˆ}$. 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.