# Canadian Mathematical Olympiad 1988 

## PROBLEM 1

For what values of $b$ do the equations: $1988 x^{2}+b x+8891=0$ and $8891 x^{2}+b x+$ $1988=0$ have a common root?

Problem 2
A house is in the shape of a triangle, perimeter $P$ metres and area $A$ square metres. The garden consists of all the land within 5 metres of the house. How much land do the garden and house together occupy?

## Problem 3

Suppose that $S$ is a finite set of at least five points in the plane; some are coloured red, the others are coloured blue. No subset of three or more similarly coloured points is collinear. Show that there is a triangle
(i) whose vertices are all the same colour, and
(ii) at least one side of the triangle does not contain a point of the opposite colour.

Problem 4
Let $x_{n+1}=4 x_{n}-x_{n-1}, x_{0}=0, x_{1}=1$, and $y_{n+1}=4 y_{n}-y_{n-1}, y_{0}=1, y_{1}=2$.
Show for all $n \geq 0$ that $y_{n}^{2}=3 x_{n}^{2}+1$.
Problem 5
Let $S=\left\{a_{1}, a_{2}, \ldots, a_{r}\right\}$ denote a sequence of integers. For each non-empty subsequence $A$ of $S$, we define $p(A)$ to be the product of all the integers in $A$. Let $m(S)$ be the arithmetic average of $p(A)$ over all non-empty subsets $A$ of $S$. If $m(S)=13$ and if $m\left(S \cup\left\{a_{r+1}\right\}\right)=49$ for some positive integer $a_{r+1}$, determine the values of $a_{1}, a_{2}, \ldots, a_{r}$ and $a_{r+1}$.