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\begin{center}
{\Large \bf THE 2001 CANADIAN MATHEMATICAL OLYMPIAD}
\end{center}

\noindent
\begin{enumerate}

\vspace*{0.5cm}
\item 
{\bf Randy:} ``Hi Rachel, that's an interesting quadratic equation you
 have written down.  What are its roots?''
 \newline
 {\bf Rachel:} ``The roots are two positive integers.  One of the roots is
 my age, and the other root is the age of my younger brother, Jimmy.''
 \newline
 {\bf Randy:} ``That is very neat!  Let me see if I can figure out how old
 you and Jimmy are.  That shouldn't be too difficult since all of your
 coefficients are integers.  By the way, I notice that the sum of the three 
 coefficients is a prime number.''
 \newline
 {\bf Rachel:} ``Interesting.  Now figure out how old I am.''
 \newline
 {\bf Randy:} ``Instead, I will guess your age and substitute it for $x$
 in your quadratic equation \dots darn, that gives me $-55$, and not $0$.''
 \newline
 {\bf Rachel:} ``Oh, leave me alone!''
 \begin{enumerate}
   \item Prove that Jimmy is two years old.
   \item Determine Rachel's age.
 \end{enumerate}

\item 
There is a board numbered $-10$ to $10$ as shown.  Each square is
 coloured either red or white, and the sum of the numbers on the red
 squares is $n$.  Maureen starts with a token on the square labeled $0$.
 She then tosses a fair coin ten times.  Every time she flips heads, she
 moves the token one square to the right. Every time she flips tails, she
 moves the token one square to the left. At the end of the ten flips, the
 probability that the token finishes on a red square is a rational number
 of the form $\frac {a}{b}$.  Given that $a+b=2001$, determine the largest
 possible value for $n$. 
 %\vspace{1ex}\newline
 \centerline{\epsfbox{img1-2001.eps}}
 

\item
Let $ABC$ be a triangle with $AC > AB$. 
 Let $P$ be the intersection point of the perpendicular bisector of $BC$ and
 the internal angle bisector of $\angle A$.  Construct points $X$ on $AB$
 (extended) and $Y$ on $AC$ such that $PX$ is perpendicular to $AB$ and
 $PY$ is perpendicular to $AC$. 
 Let $Z$ be the intersection point of $XY$ and $BC$.
 Determine the value of $BZ/ZC$.
 %\vspace{1ex}\newline
 \centerline{\epsfbox{img2-2001.eps}}


\item
Let $n$ be a positive integer.
 Nancy is given a rectangular table in which each entry is a positive
 integer.  She is permitted to make either of the following two moves:
 \begin{enumerate}
   \item  select a row and multiply each entry in this row by $n$.
   \item  select a column and subtract $n$ from each entry in this column.
 \end{enumerate}
 Find all possible values of $n$ for which the following statement is true:
 \begin{quote}
 Given any rectangular table, it is possible for Nancy to perform a finite
 sequence of moves to create a table in which each entry is~$0$.
 \end{quote}

\item
Let $P_0$, $P_1$, $P_2$ be three points on the circumference of a circle
 with radius $1$, where $P_1P_2 = t < 2$.
 For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of
 $\triangle P_{i-1} P_{i-2} P_{i-3}$.
 \begin{enumerate}
  \item
   Prove that the points $P_1, P_5, P_9, P_{13},\ldots $ are collinear.
  \item
   Let $x$ be the distance from $P_1$ to $P_{1001}$, and let $y$ be the
   distance from $P_{1001}$ to $P_{2001}$.    
   Determine all values of $t$ for which $\sqrt[500]{x/y}$ is an integer.
 \end{enumerate}

\end{enumerate}

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