# THE 1998 CANADIAN MATHEMATICAL OLYMPIAD 

1. Determine the number of real solutions $a$ to the equation

$$
\left[\frac{1}{2} a\right]+\left[\frac{1}{3} a\right]+\left[\frac{1}{5} a\right]=a
$$

Here, if $x$ is a real number, then $[x]$ denotes the greatest integer that is less than or equal to $x$.
2. Find all real numbers $x$ such that

$$
x=\left(x-\frac{1}{x}\right)^{1 / 2}+\left(1-\frac{1}{x}\right)^{1 / 2}
$$

3. Let $n$ be a natural number such that $n \geq 2$. Show that

$$
\frac{1}{n+1}\left(1+\frac{1}{3}+\cdots+\frac{1}{2 n-1}\right)>\frac{1}{n}\left(\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2 n}\right) .
$$

4. Let $A B C$ be a triangle with $\angle B A C=40^{\circ}$ and $\angle A B C=60^{\circ}$. Let $D$ and $E$ be the points lying on the sides $A C$ and $A B$, respectively, such that $\angle C B D=40^{\circ}$ and $\angle B C E=70^{\circ}$. Let $F$ be the point of intersection of the lines $B D$ and $C E$. Show that the line $A F$ is perpendicular to the line $B C$.
5. Let $m$ be a positive integer. Define the sequence $a_{0}, a_{1}, a_{2}, \ldots$ by $a_{0}=0, a_{1}=m$, and $a_{n+1}=m^{2} a_{n}-a_{n-1}$ for $n=1,2,3, \ldots$. Prove that an ordered pair $(a, b)$ of non-negative integers, with $a \leq b$, gives a solution to the equation

$$
\frac{a^{2}+b^{2}}{a b+1}=m^{2}
$$

if and only if $(a, b)$ is of the form $\left(a_{n}, a_{n+1}\right)$ for some $n \geq 0$.