# 1970 Canadian MO Problems

## Contents

## Problem 1

Find all number triples such that when any of these numbers is added to the product of the other two, the result is 2.

## Problem 2

Given a triangle with angle obtuse and with altitudes of length and as shown in the diagram, prove that . Find under what conditions .

## Problem 3

A set of balls is given. Each ball is coloured red or blue, and there is at least one of each colour. Each ball weighs either 1 pound or 2 pounds, and there is at least one of each weight. Prove that there are two balls having different weights and different colours.

## Problem 4

a) Find all positive integers with initial digit 6 such that the integer formed by deleting 6 is of the original integer.

b) Show that there is no integer such that the deletion of the first digit produces a result that is of the original integer.

## Problem 5

A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths , , and of the sides of the quadrilateral satisfy the inequalities

## Problem 6

Given three non-collinear points , construct a circle with centre such that the tangents from and are parallel.

## Problem 7

Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.

## Problem 8

Consider all line segments of length 4 with one end-point on the line and the other end-point on the line . Find the equation of the locus of the midpoints of these line segments.

## Problem 9

Let be the sum of the first terms of the sequence a) Give a formula for .

b) Prove that where and are positive integers and .

## Problem 10

Given the polynomial with integer coefficients , and given also that there exist four distinct integers , , and such that show that there is no integer such that .