# Canadian Mathematical Olympiad 

1987

## Problem 1

Find all solutions of $a^{2}+b^{2}=n$ ! for positive integers $a, b, n$ with $a \leq b$ and $n<14$.

## Problem 2

The number 1987 can be written as a three digit number $x y z$ in some base $b$. If $x+y+z=1+9+8+7$, determine all possible values of $x, y, z, b$.

Problem 3
Suppose $A B C D$ is a parallelogram and $E$ is a point between $B$ and $C$ on the line $B C$. If the triangles $D E C, B E D$ and $B A D$ are isosceles what are the possible values for the angle $D A B$ ?

Problem 4
On a large, flat field $n$ people are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When $n$ is odd show that there is at least one person left dry. Is this always true when $n$ is even?

## Problem 5

For every positive integer $n$ show that

$$
[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4 n+1}]=[\sqrt{4 n+2}]=[\sqrt{4 n+3}]
$$

where $[x]$ is the greatest integer less than or equal to $x$ (for example $[2.3]=2$, $[\pi]=3,[5]=5)$.