# Canadian Mathematical Olympiad 

1970

## Problem 1

Find all number triples $(x, y, z)$ such that when any one of these numbers is added to the product of the other two, the result is 2 .

## Problem 2

Given a triangle $A B C$ with angle $A$ obtuse and with altitudes of length $h$ and $k$ as shown in the diagram, prove that $a+h \geq b+k$. Find under what conditions $a+h=b+k$.

## 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 2 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 this 6 is $1 / 25$ of the original integer.
b) Show that there is no integer such that deletion of the first digit produces a result which is $1 / 35$ 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 $a, b, c$ and $d$ of the sides of the quadrilateral satisfy the inequalities

$$
2 \leq a^{2}+b^{2}+c^{2}+d^{2} \leq 4
$$

## Problem 6

Given three non-collinear points $A, B, C$, construct a circle with centre $C$ such that the tangents from $A$ and $B$ to the circle 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 $y=x$ and the other end-point on the line $y=2 x$. Find the equation of the locus of the midpoints of these line segments.

## PROblem 9

Let $f(n)$ be the sum of the first $n$ terms of the sequence

$$
0,1,1,2,2,3,3,4,4,5,5,6,6, \ldots
$$

a) Give a formula for $f(n)$.
b) Prove that $f(s+t)-f(s-t)=s t$ where $s$ and $t$ are positive integers and $s>t$.

Problem 10
Given the polynomial

$$
f(x)=x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\cdots+a_{n-1} x+a_{n}
$$

with integral coefficients $a_{1}, a_{2}, \ldots, a_{n}$, and given also that there exist four distinct integers $a, b, c$ and $d$ such that

$$
f(a)=f(b)=f(c)=f(d)=5
$$

show that there is no integer $k$ such that $f(k)=8$.