# 1969 Canadian MO Problems

## Contents

## Problem 1

Show that if and are not all zero, then for every positive integer

## Problem 2

Determine which of the two numbers , is greater for any .

## Problem 3

Let be the length of the hypotenuse of a right triangle whose two other sides have lengths and . Prove that . When does the equality hold?

## Problem 4

Let be an equilateral triangle, and be an arbitrary point within the triangle. Perpendiculars are drawn to the three sides of the triangle. Show that, no matter where is chosen, .

## Problem 5

Let be a triangle with sides of length , and . Let the bisector of the cut at . Prove that the length of is

## Problem 6

Find the sum of , where .

## Problem 7

Show that there are no integers for which .

## Problem 8

Let be a function with the following properties:

1) is defined for every positive integer ;

2) is an integer;

3) ;

4) for all and ;

5) whenever .

Prove that .

## Problem 9

Show that for any quadrilateral inscribed in a circle of radius the length of the shortest side is less than or equal to .

## Problem 10

Let be the right-angled isosceles triangle whose equal sides have length 1. is a point on the hypotenuse, and the feet of the perpendiculars from to the other sides are and . Consider the areas of the triangles and , and the area of the rectangle . Prove that regardless of how is chosen, the largest of these three areas is at least .