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\begin{center}
{\Large \bf THE 1998 CANADIAN MATHEMATICAL OLYMPIAD }
\end{center}
\noindent
\begin{enumerate}

\vspace*{0.5cm}
\item
Determine the number of real solutions $a$ to the equation
\begin{displaymath}
\left[ \,\frac{1}{2}\;a \,\right] + \left[ \,\frac{1}{3}\;a\, \right] + \left[
 \, \frac{1}{5}\;a \,\right] = a\;.
\end{displaymath}

Here, if $x$ is a real number, then $[\,x\,]$ denotes the greatest integer that 
is less than or equal to $x$.

\vspace*{0.5cm}
\item
 Find all real numbers $x$ such that
\begin{displaymath}
x = \left( x - \frac{1}{x} \right)^{1/2} + \left( 1 - \frac{1}{x} \right)^{1/2}.
\end{displaymath}

\vspace*{0.5cm}
\item
 Let $n$ be a natural number such that $n \geq 2$.  Show that
\begin{displaymath}
\frac{1}{n+1} \left( 1 + \frac{1}{3} + \cdot \cdot \cdot +
\frac{1}{2n-1} \right) > \frac{1}{n} \left( \frac{1}{2} + \frac{1}{4}
+ \cdot \cdot \cdot + \frac{1}{2n} \right).
\end{displaymath}

\vspace*{0.5cm}
\item
Let $ABC$ be a triangle with $\angle BAC = 40^{\circ}$ and $\angle ABC =
60^{\circ}$.  Let
$D$ and $E$ be the points lying on the sides $AC$ and $AB$, respectively,
such that
$\angle CBD = 40^{\circ}$ and $\angle BCE = 70^{\circ}$.  Let $F$ be the
point 
of intersection of the lines $BD$ and $CE$.  Show that 
%where the lines $BD$ and $CE$ intersect.  Show that 
the line $AF$ is perpendicular to the line $BC$.

\vspace*{0.5cm}
\item
Let $m$ be a positive integer. Define the sequence $a_0, a_1, a_2,
\ldots$ by $a_0 = 0,\; a_1 = m,$ % \linebreak 
and $a_{n+1} = m^2a_n - a_{n-1}$ 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
$${\displaystyle \frac{a^2 + b^2}{ab + 1} = m^2}$$
if and only if $(a,b)$ is of the form $(a_n,a_{n+1})$ for
some $n \geq 0$.
\end{enumerate}

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