Solutions should be submitted to

Dr. Valeria Pandelieva

641 Kirkwood Avenue

Ottawa, ON K1Z 5X5

no later than **May 31, 2001**, and no sooner than *May 21, 2001*.

- 73.
- Solve the equation:

- 74.
- Prove that among any group of $n+2$ natural numbers, there can be found two numbers so that their sum or their difference is divisible by $2n$.

- 75.
- Three consecutive natural numbers, larger than 3, represent the lengths of the sides of a triangle. The area of the triangle is also a natural number.

- (a) Prove that one of the altitudes ``cuts'' the triangle into two triangles, whose side lengths are natural numbers.

- (b) The altitude identified in (a) divides the side which is perpendicular to it into two segments. Find the difference between the lengths of these segments.

- 76.
- Solve the system of equations:

(The logarithms are taken to base 10.)

- 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 less than $\sqrt{2}$. What is the largest natural number $n$ for which this is possible?

- 78.
- A truck travelled from town $A$ to town $B$ over several days. During the first day, it covered $1/n$ of the total distance, where $n$ is a natural number. During the second day, it travelled $1/m$ of the remaining distance, where $m$ is a natural number. During the third day, it travelled $1/n$ of the distance remaining after the second day, and during the fourth day, $1/m$ of the distance remaining after the third day. Find the values of $m$ and $n$ if it is known that, by the end of the fourth day, the truck had travelled $3/4$ of the distance between $A$ and $B$. (Without loss of generality, assume that $m<n$.)