# 1971 Canadian MO Problems

## Contents

## Problem 1

is a chord of a circle such that and Let be the center of the circle. Join and extend to cut the circle at Given find the radius of the circle

## Problem 2

Let and be positive real numbers such that . Show that .

## Problem 3

is a quadrilateral with . If is greater than , prove that .

## Problem 4

Determine all real numbers such that the two polynomials and have at least one root in common.

## Problem 5

Let , where the coefficients are integers. If and are both odd, show that has no integral roots.

## Problem 6

Show that, for all integers , is not a multiple of 121.

## Problem 7

Let be a five digit number (whose first digit is non-zero) and let be the four digit number formed from n by removing its middle digit. Determine all such that is an integer.

## Problem 8

A regular pentagon is inscribed in a circle of radius . is any point inside the pentagon. Perpendiculars are dropped from to the sides, or the sides produced, of the pentagon.

a) Prove that the sum of the lengths of these perpendiculars is constant.

b) Express this constant in terms of the radius .

## Problem 9

Two flag poles of height and are situated units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.

## Problem 10

Suppose that people each know exactly one piece of information, and all pieces are different. Every time person phones person , tells everything that knows, while tells nothing. What is the minimum number of phone calls between pairs of people needed for everyone to know everything? Prove your answer is a minimum.