# Canadian Mathematical Olympiad 

1985

## Problem 1

The lengths of the sides of a triangle are 6,8 and 10 units. Prove that there is exactly one straight line which simultaneously bisects the area and perimeter of the triangle.

Problem 2
Prove or disprove that there exists an integer which is doubled when the initial digit is transferred to the end.

## PROBLEM 3

Let $P_{1}$ and $P_{2}$ be regular polygons of 1985 sides and perimeters $x$ and $y$ respectively. Each side of $P_{1}$ is tangent to a given circle of circumference $c$ and this circle passes through each vertex of $P_{2}$. Prove $x+y \geq 2 c$. (You may assume that $\tan \theta \geq \theta$ for $\left.0 \leq \theta<\frac{\pi}{2}\right)$.

## Problem 4

Prove that $2^{n-1}$ divides $n$ ! if and only if $n=2^{k-1}$ for some positive integer $k$.
Problem 5
Let $1<x_{1}<2$ and, for $n=1,2, \ldots$, define $x_{n+1}=1+x_{n}-\frac{1}{2} x_{n}^{2}$. Prove that, for $n \geq 3,\left|x_{n}-\sqrt{2}\right|<2^{-n}$.