\documentclass[12pt]{amsart}

\usepackage{graphics}

\title{Department of Mathematics\\
Competition for the Auman Prize\\
2000}

\everymath{\displaystyle}
\setlength{\parindent}{0pt}

\begin{document}

\maketitle

\thispagestyle{empty}

Any student enrolled in MATH 166, Honors Calculus II, may enter.
Each competitor must solve the problems independently under the honor
code.  At least two problems must be solved to qualify. The student who
provides the most complete solutions will win a cash prize of~\$200.
Solutions must be submitted to Prof.~Dennis Snow, CCMB 313, by noon
Friday, March~24.

\bigskip

\begin{enumerate}
\item
Let $f(x)$ be differentiable at $x=a$, and $f(a)\ne0$. Evaluate
\[ \lim_{n\to\infty}\left[\frac{f(a+1/n)}{f(a)}\right]^n \]

\bigskip
\item
Prove that there is no continuous function $f$ with domain the closed
interval $[0,1]$ and range the open interval $(0,1)$.

\bigskip
\item
Calculate $\lim_{x\to\infty}x\int_0^x e^{t^2-x^2}dt$.

\bigskip
\item
Determine how far a stack of $n$ identical books can be made to
project over the edge of a table by stacking them one on top of the
other. Determine the theoretical limit of this overhang as $n\to\infty$.

\[
\resizebox{!}{0.75in}{\includegraphics{auman00.eps}}
\]

\bigskip
\item Show if there are $n\ge2$ people in a room, then two of them know
the same number of people among those present. (Assume that if $A$ knows
$B$, then $B$ knows $A$).

\end{enumerate}

\end{document}
