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\begin{center} {\large \bf Mathcamp 2002 Application Quiz} \end{center}

\begin{center}
{\bf Instructions}
\end{center}

Try to solve as many of these problems as you can.  We certainly don't
expect every applicant to solve every problem on the quiz -- we often
accept people who can do only five, or even fewer -- but we do want you
to give all the questions some thought. Some of them may be tougher
than you're used to. Keep turning them over in your head: often the
crucial insight comes when you least expect it!  

You are welcome to use outside resources, such as books or the Web, as
long as you acknowledge and reference them. Of course, all the actual 
work
must be your own.  Remember that your reasoning matters more to us than
your final answers, so justify all your assertions, and don't hesitate 
to 
submit partial solutions.

Good luck, and have fun!

\begin{center}
{\bf Problems}
\end{center}

\begin{enumerate}

\item In a certain 
game, there are three piles of stones: one with $22$
stones, one with $14$ stones, and one with $12$ stones. At each turn, you 
can
double the number of stones in a pile by transferring stones to it from
one other pile. (For example, your first move can be to transfer $12$ 
stones
from the first pile to the third pile.) The game ends if the three piles
are equal in size. Can you find a way to end the game in three moves?

\item Imagine a corridor with $2002$ light switches. Initially, all the 
lights
are off. Then $2002$ people walk through the corridor, one after the 
other.
The first person flips every switch, the second person flips every other
switch (starting with the second), the third person flips every third
switch (starting with the third), and so on. After all the people have 
gone through, which lights are on and which are off?

\item Show that you cannot cover a circular disk with two circular disks 
of smaller diameter.

\item Three bugs are crawling on the coordinate plane.  They move one at a
time, and each bug will only crawl in a direction parallel to the line 
joining the
other two.

\begin{enumerate}

\item[(a)] If the bugs start out at $(0,0)$, $(3,0)$, and $(0,3)$, is 
it 
possible 
that
after some time the first bug will end up back where it started, while the
other two bugs switch places?

\item[(b)] Can the three bugs end up at $(1,2)$, $(2,5)$, and $(-2,3)$?

\end{enumerate}

\item  Suppose $15$ pennies are arranged in a triangle, as in the 
following 
diagram:

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Show that inside this triangle, there is guaranteed to be an equilaterial 
triangle in which a