\documentclass[a4paper,12pt,twoside]{amsart}
\title{Math 304 examination, Tuesday, March 30, 2010}
\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{Tuesday, March 30, 2010}
\usepackage{hfoldsty}
\usepackage{typearea}
\theoremstyle{definition}
\newtheorem{problem}{Problem}
\renewcommand{\theenumi}{\alph{enumi}}
\renewcommand{\labelenumi}{(\theenumi).}
\begin{document}
\maketitle
\thispagestyle{empty}
You may use modern notation in your work; but Problems~\ref{prob:quad} and~\ref{prob:OK} should involve diagrams.
\begin{problem}
A straight line is cut into equal and unequal segments. What is the
relationship between the square on the half and the rectangle
contained by the unequal segments?
\end{problem}
\vfill
\begin{problem}\label{prob:quad}
A square is equal to three roots and twenty-eight dirhams.
What is the root?
Give a geometrical justification of your answer (as Mu\d
hammad ibn M\=us\=a al-Khw\=arizm\=\i{} or Th\=abit ibn Qurra did).
\end{problem}
\vfill\vfill
\newpage
\begin{problem}\label{prob:OK}
Suppose a cube and nine sides are equal to ten.
Find the side by taking the
intersection of two conic sections (as Omar Khayy\=am did). It is preferable if one of those
sections is a circle.
\end{problem}
\newpage
\begin{problem}
Again, a cube and nine sides are equal to ten.
\begin{enumerate}
\item
Find the side numerically, as the difference of the cube roots of a
\emph{binomium} and an \emph{apotome,} by Cardano's method
(really Tartaglia's
method); your steps should be clearly justifiable.
\item
The side is in fact a whole
number; which one?
\end{enumerate}
\end{problem}
\newpage
\begin{problem}
A square-square, twelve squares, and thirty-six are equal to
seventy-two sides. In finding the side by Cardano's method (really
Ferrari's method), you first solve a cubic equation.
\begin{enumerate}
\item\label{part:1}
Obtain that cubic equation. %in the form ``cube and squares equal to
%number''.
\item
Convert that cubic equation to an equation of the form ``cube equal to roots and
number''.
\item
The cubic equation in~\eqref{part:1} should have $6$ as a root. Use this to
find the side in the original fourth-degree equation.
\end{enumerate}
\end{problem}
\end{document}