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\begin{center} {\large \bf Mathcamp 2005 Application Quiz} \end{center}

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{\bf Instructions}
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We call it a quiz, but it’s really a challenge: a way for you to
show us how you approach new problems and new topics in
mathematics. What matters to us are not only your final results,
but your reasoning. In every problem, be sure to justify your
assertions, explain how you got your answer, and convince us that
your solution is correct!

We certainly don't expect every applicant to solve every problem.
In the past, we have often admitted applicants who could do only
five, sometimes even fewer. However, you should definitely try all
the problems, and send us the results of your efforts: partial
solutions, conjectures, methods— everything counts. Sometimes it
may take a while to find the right way of approaching a problem.
Be patient—there is no time limit on this quiz.

You are welcome to consult books or the Web, as long as you
acknowledge and provide references for your sources. If you need
clarification on a problem, you can send email to {\sf
quiz@mathcamp.org}, but you may not consult or get hints from
anyone else. None of the problems require a computer -- they are
all based on insight. You are welcome to use a computer if you
like, but first read a word of warning at {\sf
http://www.mathcamp.org/computers.php}.

For some tips on writing proofs, see {\sf
http://www.mathcamp.org/proofs.html}.

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Good luck, and have fun!

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{\bf Problems}
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