# Canadian Mathematical Olympiad 

1972

## Problem 1

Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.

## PROblem 2

Let $a_{1}, a_{2}, \ldots, a_{n}$ be non-negative real numbers. Define $M$ to be the sum of all products of pairs $a_{i} a_{j}(i<j)$, i.e.,

$$
M=a_{1}\left(a_{2}+a_{3}+\cdots+a_{n}\right)+a_{2}\left(a_{3}+a_{4}+\cdots+a_{n}\right)+\cdots+a_{n-1} a_{n}
$$

Prove that the square of at least one of the numbers $a_{1}, a_{2}, \ldots, a_{n}$ does not exceed $2 M / n(n-1)$.

## Problem 3

a) Prove that 10201 is composite in any base greater than 2.
b) Prove that 10101 is composite in any base.

## PROBLEM 4

Describe a construction of a quadrilateral $A B C D$ given:
(i) the lengths of all four sides;
(ii) that $A B$ and $C D$ are parallel;
(iii) that $B C$ and $D A$ do not intersect.

## PROBLEM 5

Prove that the equation $x^{3}+11^{3}=y^{3}$ has no solution in positive integers $x$ and $y$.

## PROblem 6

Let $a$ and $b$ be distinct real numbers. Prove that there exist integers $m$ and $n$ such that $a m+b n<0, b m+a n>0$.

## PROBLEM 7

a) Prove that the values of $x$ for which $x=\left(x^{2}+1\right) / 198$ lie between $1 / 198$ and 197.99494949 .
b) Use the result of a) to prove that $\sqrt{2}<1.41 \overline{421356}$.
c) Is it true that $\sqrt{2}<1.41421356$ ?

## Problem 8

During a certain election campaign, $p$ different kinds of promises are made by the various political parties $(p>0)$. While several parties may make the same promise, any two parties have at least one promise in common; no two parties have exactly the same set of promises. Prove that there are no more than $2^{p-1}$ parties.

Problem 9
Four distinct lines $L_{1}, L_{2}, L_{3}, L_{4}$ are given in the plane: $L_{1}$ and $L_{2}$ are respectively parallel to $L_{3}$ and $L_{4}$. Find the locus of a point moving so that the sum of its perpendicular distances from the four lines is constant.

Problem 10
What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?