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\begin{center} {\large \bf Mathcamp 2001 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, sojustify 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: the first pile has
   $22$ stones, the second pile has $14$ stones, and the third pile has
$12$
   stones.  At each turn, a player may double the number of stones in a
   pile by transferring stones to it from one other pile.  (For example,
   one could move $12$ stones from the first pile to the third pile.)  The
   game ends if the three piles all have the same size.  Can you find a
   way to end the game in three moves?

\item Suppose you randomly pick two different integers between $1$ and
   $1,000,000$.  Is their sum more likely to be even or odd?

\item Show that there are no positive integers $m$ and $n$ such that
$m(m+1) =
   n(n+2)$.

\item Five mathematically gifted pirates, named Angry, Boorish, Crummy,
   Dirty, and Evil, must divide a loot of $100$ gold doubloons.  According
   to pirate law, Angry, the leader, can distribute the coins as she sees
   fit.  However, when she is done, all five pirates (including Angry 
   herself) take a vote: if more than half of them are unhappy with the
   distribution, Angry has to walk the plank (leaving her share of gold
   behind). The remaining four pirates, with Boorish as their new leader,
   then repeat the process: Boorish divides the loot, they all vote, and
   if more than half are unhappy then Boorish is done away with and
   Crummy takes charge of the gang. This process continues until
   the treasure is successfully distributed.

\begin{enumerate}
    \item How should Angry divide the money to ensure herself as   
        large a share of the loot as possible?  (You may assume that
        every pirate completely understands the strategy behind this
        process, and will always vote in such a way as to maximize his
        or her share.  Furthermore, each pirate is bloodthirsty, and
        will vote to kill the current leader if it won't decrease his
        or her ultimate share of the loot or chances of survival.)

    \item What happens when there are more than five pirates?  More than
two
        hundred pirates?
\end{enumerate}

\item Aytek the Ant is standing at the point $(0,10)$ in the x-y plane.
   Aytek is going to travel to the point $(20,14)$, but in doing so, he
   must first travel to the x-axis -- where much-needed sugar can be
   found -- and walk along it for exactly $10$ units.  What route should
   Aytek take, in order to complete the whole trip from $(0,10)$ to
   $(20,14)$ in as short a distance as possible?

\item If $S_n$ denotes the sum of the decimal digits of the number $2^n$,
 either
   show that $S_n$ does not equal $S_{n+1}$ for all positive integers $n$,
or
   find a counterexample.

\item The decimal expansion of the fraction $1/11 = .090909\ldots$ 
 consists
   of the two digits $09$ repeating over and over; we say that this
   decimal expansion has period length $2$.  Similarly, the decimal  
   expansion of $1/37 = .027027027\ldots$ has period length $3$.  Can you
   find other integers $n$ such that the decimal expansion of $1/n$ has
period
   length $2$?  period length $3$?  Can you find all prime numbers $p$ 
such
that 
   the decimal expansion of $1/p$ has period length at most $6$?  Can you
   find any prime numbers $p$ so that the octal (i.e., base $8$) expansion
  of
   $1/p$ has period length $3$?

\item Suppose $a$, $b$, $c$, and $d$ are the side lengths of a
quadrilateral
with
   area $S$.

\begin{enumerate}
\item  Show that $2S \le ab + cd$.
\item  Show that $2S \le ac + bd$.
\item  If either one of these two inequalities is in fact an equality,
        show that the quadrilateral can be inscribed in a circle.
\end{enumerate}

\item Suppose $n$ disks, black on one side and white on the other, are
laid
   out in a straight line with a random arrangement of black sides up. You
   are playing a game of solitaire, in which a turn consists of removing a
   black disk and flipping over its immediate neighbors, if any. (Two
   disks are {\em not} considered immediate neighbors if there used to be
   a disk between them that is now gone.)  You win if you succeed in
   removing all $n$ disks.  Describe all initial configurations of disks
   that are winnable, explain how to win them, and show that all other
   configurations are unwinnable.

\item Inside a $1$ m by $1$ m square box in the xy-plane, there are
finitely
   many line segments, whose lengths sum to exactly $10$ m.  Show that
   there exists a straight line in the plane which crosses at least
   six of these line segments.  (Hint: first, show that there exists
   a straight line in the plane which crosses at least five of these
   line segments.)

\end{enumerate}

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