\documentclass [12pt]{article}
\title{A Maths Handout}
\author{J. Kinsella} \date{\today}
\begin{document}
\maketitle
%\subparagraph{e}
\begin{description}
\item [Pigeon-hole Principle] If $n+1$ objects are inserted into $n$
boxes, one of the boxes must contain at least 2 objects.
\item [Generalisation:] If $nk + 1$ objects are distributed among $n$
boxes, one of the boxes must contain $k + 1$ objects.
\end{description}
{\Large Problems}
\begin{enumerate}
\item Five points are located in the interior of a square of side 2.
Prove that there are two whose distance apart is less than $\sqrt{2}$ .
\item Prove that of any 10 points chosen inside an equilateral
triangle of side 1 there are two whose distance apart is at most $1/3$ .
\item For all positive integers $k$, prove that at least one of the
integers in the set $S = \{ 2^1 - 1,2^2 - 1,2^3 - 1,....,2^{2k} - 1 \}$
is divisible by $2k+1$ .
\item Prove that there exist integers $a ,b ,c$ not all zero and each
of absolute value less than one million such that
$|a + b \sqrt{2} + c \sqrt{3}| < 10^{-11}$
\item Study the following formula.\label{prob:formula}
\begin{equation}
\label{eq:formula}
\int_a^b f(x) dx
\end{equation}
\item Given any set of ten integers between 1 and 99 inclusive,
prove that there are two disjoint non-empty subsets of the set with
equal sums of their elements.
\item Study the following formula.\label{prob:formula2}
\begin{equation}
\label{eq:formula2}
\sum_{i=1}^N i^2=N (N+1) (2N+1)/6
\end{equation}
\item Show that given any 17 numbers it is possible to choose 5
whose sum is divisible by 5.
\item A student has 37 days left to study for an exam. She knows
that she will require no more than 60 hours of study. She also
wishes to study at least 1 hour per day. Show that no matter
how she organises her study there is a succession of days
during which she studies exactly 13 hours.
\item Can you remember the formula Eq~\ref{eq:formula} in Question~\ref{prob:formula} ?
\item Which formula did you prefer, Eq~\ref{eq:formula} in Question~\ref{prob:formula} or Eq~\ref{eq:formula2} in Question~\ref{prob:formula2}?
\end{enumerate}
\end{document}