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\begin{center} {\large \bf Mathcamp 2000 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.

The project at the end of the quiz is optional; if it intrigues you, try
it.  We'd like to see what you come up with!

Good luck, and have fun!

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

\begin{enumerate}

\item
Imagine a corridor with $100$ light switches. $100$ people walk through
the corridor one after another. 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),
etc. At the end, which switches are on and which are off?

\item Find the smallest positive integer $n$ such that ${n\over 2}$ is the
square of an integer, ${n\over 5}$ is the fifth power of an integer,
${n\over 8}$ is the eighth power of an integer, and ${n\over 11}$ is the
eleventh power of an integer.

\item
A triangle has sides of length $a$, $b$ and $c$, which satisfy the
equation $1/(a+b) + 1/(a+c) = 3/(a+b+c)$. Determine the angle opposite
the side of length $a$.

\item Yuliya rolls a pair of fair dice four times.  What is the
probability that she rolls at least a $10$, at least two times?

\item
Do there exist $1,000,000$ consecutive composite numbers?  If not, why
not?  If so, do you have any ideas on how to find them?

\item
Find all integers $x$, $y$, and $n$ such that $xy^2(x+y^2)= 2^n$.

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

\begin{enumerate}
\item 
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 switch places?

\item
Can the three bugs end up at (1,2), (2,5), and (-2,3)?
\end{enumerate}

\item
You are a secret agent assigned to carry out classified research on
the $C_n$-tower, an ultra-modern office block $n$ stories tall. Your
mission: to determine the highest floor of the building from which a
dropped egg will survive falling to the ground.  Your equipment: a
supply of exactly $k$ eggs.

\begin{enumerate}

\item
Suppose $k=1$: you have only one egg (and once it smashes, that's
it). How can you determine the highest floor that is safe? Are there
any other strategies?

\item
Suppose $k=2$. You need to complete your mission as quickly as
possible.  What strategy should you use to determine the safe height
efficiently?  (Here ``efficiently'' means that the {\em worst case
scenario} takes as few test drops as possible.)

\item
Can you find an answer for higher values of $k$?

\end{enumerate}

\item  Let $a_0 = 1$ and $a_n = \frac{\sqrt{1+a_{n-1}^2} - 1}{a_{n-1}}$.
Show that $a_n > \frac{\pi}{2^{n+2}}$.

\item Let $S_1$, $S_2$, and $S_3$ be three spheres in three-dimensional
space whose centres are not collinear. Let $k \le 8$ be the number of
planes which are tangent to all three spheres.  Let $A_i$, $B_i$, and
$C_i$ be the point of tangency between the $i^{th}$ such tangent plane ($1
\le i \le k$) and $S_1$, $S_2$, and $S_3$ respectively, and let $O_i$ be
the circumcentre of triangle $A_i B_i C_i$. Prove that the $O_i$ are all
collinear.  (If $k=0$, this statement is vacuously true.)

\end{enumerate}

\begin{center}
{\bf Optional Project}
\end{center}

\begin{enumerate}

\item[P.] 
\noindent What is the probability that a random chord on a circle is
longer than the side of an inscribed equilateral triangle?

\begin{enumerate} 
\item
Can you give mathematical arguments for 3 (or even more) {\em
different} answers!?  Is there a correct solution?  What do you think
might be the source of the paradox?
    
\item
Suppose we try to resolve the paradox experimentally: draw a
circle on the floor, drop 100 straws on it, and calculate what
percentage of them cuts a chord that is longer than the side of an
inscribed equilateral triangle. (Some straws might miss the circle
entirely; let's ignore those, since they don't cut any chord at all.)
Which of the numbers from part (a) do you think you will get, and why?
(If you wish, you can try doing the experiment and reporting on your
results!)

\end{enumerate}
\end{enumerate}

\end{document}

