\documentclass[a4paper,12pt,twoside]{amsart}
%\documentclass[a4paper,12pt,twoside]{article}

\title{Math 303 examination}
\author{David Pierce}
%\email{dpierce@metu.edu.tr}
%\address{Mathematics Dept., Middle East Technical University, Ankara 06531, Turkey}
%\urladdr{metu.edu.tr/~dpierce/Courses/303/}
 
\date{Friday, November 6, 2009}

\usepackage[polutonikogreek,english]{babel}
\usepackage[oxonia]{psgreek}

\usepackage{hfoldsty}
\usepackage{typearea}
%\usepackage{parskip}
\usepackage{paralist}
\newcommand{\Gk}[1]{\selectlanguage{polutonikogreek}#1\selectlanguage{english}}

\theoremstyle{definition}
\newtheorem{problem}{Problem}
\renewcommand{\arraystretch}{2}
\renewcommand{\labelenumi}{\theenumi.}

\begin{document}
\maketitle
\begin{problem}

  What is wrong with the following proof that all triangles are isosceles?---:
  \begin{enumerate}
  \item
  Let a triangle be given, namely $ABC$.
  \item
  Let $BC$ be bisected at $D$.
  \item
  Let a straight line, $DE$, be drawn at right angles to $BC$.
  \item
  Let also the straight line $AE$ bisect the angle $BAC$.
  \item
  Let the straight lines $BE$ and $CE$ be drawn.
  \item
  $BE=CE$.
  \item
  Let the straight line $EF$ be drawn perpendicular to $AB$.
  \item
  Let the straight line $EG$ be drawn perpendicular to $AC$.
   \item
  $AF=AG$ and
  $EF=EG$.
  \item
  $BF=CG$.
  \item
  $AF+FB=AG+GC$.
  \item
  $AF+FB=AB$ and $AG+GC=AC$.
  \item
  $AB=AC$; in particular, $ABC$ is isosceles.
  \end{enumerate}
\mbox{}\hfill  
\setlength{\unitlength}{1.5cm}
  \begin{picture}(2,0)(-1,-2)
  \put(-1,0){\line(1,0)2}
  \put(-1,0){\line(1,2){1}}
   \put(1,0){\line(-1,2){1}}
   \put(0,0){\line(0,1){2}}
   \put(-1,0){\line(2,1){1.6}}
   \put(1,0){\line(-2,1){1.6}}
   \put(0,2.05){\makebox(0,0)[b]{$A$}}
   \put(1,0){\makebox(0,0)[tl]{$C$}}
   \put(-1,0){\makebox(0,0)[tr]{$B$}}
   \put(0,-0.05){\makebox(0,0)[t]{$D$}}
   \put(0,0.6){$E$}
   \put(0.6,0.8){\makebox(0,0)[bl]{$G$}}
   \put(-0.6,0.8){\makebox(0,0)[br]{$F$}}
  \end{picture}
\end{problem}
\vfill
\begin{problem}
Write English translations of the following words:
\begin{center}
\begin{inparaenum}[(a)]
\item
\Gk{je'wrhma},
\item
\Gk{pr'oblhma},
\item
\Gk{>an'alusic},
\item
\Gk{sunj'esic},
\item
\Gk{pol'ugwnon},
\item
\Gk{tr'igwnon}.
\end{inparaenum}
\end{center}
\end{problem}
\vfill
\begin{problem}
Write the letters of the Greek alphabet in the standard
order.  Give only the capital letters \emph{or} only the minuscule
letters.  
\begin{center}

\begin{tabular}{|*{12}{c|}}\hline
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}\\\hline\hline
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}&
\phantom{W}\\\hline
\end{tabular}
\end{center}
\end{problem}

\newpage
\begin{problem}
Proclus writes:
\begin{quote}
  Every problem and every theorem that is furnished with all its parts
  should contain the following elements: 
  \begin{enumerate}\renewcommand{\labelenumi}{(\theenumi)}
  \item
  an \emph{enunciation} (\Gk{pr'otasic}), 
  \item
  an \emph{exposition} (or \emph{setting out:} \Gk{>'ekjesic}), 
  \item
  a \emph{specification} (or \emph{definition of goal:} \Gk{diorism'oc}), 
  \item
  a \emph{construction} (\Gk{kataskeu'h}), 
  \item
  a \emph{proof} (\Gk{>ap'o\-deixic}), and 
  \item
  a \emph{conclusion} (\Gk{sump'erasma}). 
  \end{enumerate}
  \end{quote}
  Below is the enunciation (in Heath's translation) of Proposition I.6
  of Euclid's \emph{Elements.}  Supply the
  remaining parts (in your own words, which may or may not be
  Euclid's). 
  \begin{quote}
  \emph{If in a triangle two angles be equal to one another, the sides which
  subtend the equal angles will also be equal to one another.}
  \end{quote}
 \end{problem}
  \newpage
  \begin{problem}
  \emph{Without} using Euclid's method of `application', prove
  Proposition I.8 of the \emph{Elements,} whose enunciation is,
  \begin{quote}
  \emph{If two triangles have two sides equal to two sides
    respectively, and have also the base equal to the base, they will
    also have the angles equal which are contained by the equal
    straight lines.}
  \end{quote}
  \end{problem}
  \newpage
  \begin{problem}
  In triangle $ABC$, suppose $BC$ is bisected at $D$, and straight line $AD$ is drawn.  Assuming $AB$ is greater than $AC$, prove that angle $BAD$ is less than $DAC$.
  \end{problem}
  \vfill
\end{document}
