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\begin{center}
{\Large \bf THE 2000 CANADIAN MATHEMATICAL OLYMPIAD}
\end{center}

\begin{enumerate}

\item 
At 12:00 noon, Anne, Beth and Carmen begin running laps around
a circular track of length three hundred meters,
all starting from the same point on the track.
Each jogger maintains a constant speed in one of the two possible
directions for an indefinite period of time.
Show that if Anne's speed is different from the other two speeds,
then at some later time Anne will be at least one hundred
meters from each of the other runners.  (Here, distance is measured along
the shorter of the two arcs separating two runners.)

\item 
A {\it permutation\/} of the integers $1901, 1902, \ldots, 2000$
is a sequence  $a_1, a_2, \ldots, a_{100}$ in which each
of those integers appears exactly once.
Given such a permutation, we form the sequence of partial sums  
\begin{displaymath}
s_1 = a_1,\;\;s_2 = a_1 + a_2,\;\;s_3 = a_1 + a_2 + a_3,
\; \ldots\;, \; s_{100} = a_1 + a_2 + \cdots + a_{100}.
\end{displaymath}
How many of these permutations will have no terms of the sequence
$s_1, \ldots, s_{100}$ divisible by three?
 

\item
Let $A = (a_1, a_2, \ldots ,a_{2000})$ be a sequence of integers
each lying in the interval $[-1000,1000]$.
Suppose that the entries in A sum to $1$.
Show that some nonempty subsequence of $A$ sums to zero.


\item
Let $ABCD$ be a convex quadrilateral with
\begin{eqnarray*}
\angle CBD & = & 2 \angle ADB,\\
\angle ABD & = & 2 \angle CDB\\
\mbox{and} \hspace*{1.5cm} AB & = &CB.
\end{eqnarray*}
Prove that $AD = CD$.

\item
\rule{-0.7ex}{0ex}
%(Goddyn modification of a 1996 problem)
%\marginpar %{\hfill [5]}
 Suppose that the real numbers $a_1, a_2, \ldots, a_{100}$ satisfy
\begin{eqnarray*}
a_1 \ge a_2 \ge \cdots \ge a_{100}                &\!\!\!\ge\!\!\!& 0,  \\
a_1+a_2                                           &\!\!\!\le\!\!\!& 100 \\
\mbox{and} \hspace*{1.0cm} a_3+a_4+\cdots+a_{100} &\!\!\!\le\!\!\!& 100.
\end{eqnarray*}
Determine the maximum possible value of
$a_1^2 + a_2^2 + \cdots + a_{100}^2$,
and find all possible sequences
$a_1, a_2, \ldots , a_{100}$
which achieve this maximum.


\end{enumerate}

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