\documentclass[12pt,HW250,ifthen]{article}
\usepackage{comment}
\newcommand{\into}{{\rightarrow}}
\newcommand{\lf}{\left\lfloor}
\newcommand{\rf}{\right\rfloor}
\newcommand{\lc}{\left\lceil}
\newcommand{\rc}{\right\rceil}
\newcommand{\Ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
\newcommand{\Z}{{\sf Z}}
\newcommand{\N}{{\sf N}}
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\newcommand{\R}{{\sf R}}
\newcommand{\Rpos}{{\sf R}^+}
\usepackage{amsmath}

\begin{document}

\centerline{Homework 8, Morally due Mon Apr 16, 3:30PM}

\newif{\ifshowsoln}
\showsolntrue % comment out to NOT show solution inside \ifshowsoln and \fi blocks.

Throughout this HW:

\begin{itemize}
\item
Let $f(m,s)$ be the muffin function (from the talk Bill gave on Muffins).
\item
To prove that, say $f(11,5)=\frac{13}{30}$ you would need to BOTH
give a PROCEDURE that allocates 11 muffins to 5 people with smallest piece $\frac{13}{30}$
AND prove that there is no BETTER procedure.
\item
You CANNOT use the Floor-Ceiling Theorem, though you can use the same kind of reasoning in
a particular case.
\end{itemize}

\begin{enumerate}

\item
(50 points)
\newcommand{\ob}[1]{\frac{#1}{5}}
Prove $f(9,5)=\ob 2$.

\item
(50 points)
Prove $f(7,6)=\frac{1}{3}$.

\end{enumerate}

\end{document}