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\centerline{\bf Homework 9, MORALLY Due 10:00AM April 22} 
\newcommand{\SPS}{{\rm SPS}}
\newcommand{\IO}{\exists^\infty}
\newcommand{\st}{\mathrel{:}}
\newcommand{\goes}{\Rightarrow}
\newcommand{\NSQ}{{\rm NSQ}}
\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}}
\newcommand{\Q}{{\sf Q}}
\newcommand{\R}{{\sf R}}
\newcommand{\D}{{\sf D}}
\newcommand{\Rpos}{{\sf R}^+}

\begin{enumerate}


\item
(30 points) 
Recall the BEE sequence. 

\smallskip
$a_1=1$

\smallskip
$(\forall n\ge 2)[a_n = a_{n-1} + a_{\floor{n/2}}]$


Try to prove the following:


{\it There are an infinite number of $n$ such that $a_n\equiv 0 \pmod 9$.}

You will do this by following the proof for moe 7.
Show where the proof breaks down. 

\smallskip

(Recall that the the statement $(\IO n)[a_n\equiv 0 \pmod 9]$ is not known to be true or false,
though empirical evidece suggests that its true.  Our approach did not work; however, some
other approach might.)


\newpage

\item 
(30 points) 
Recall that $|\SPS(1,\ldots,n)|=\frac{n(n+1)}{2}+1$. 

Assume $n$ is large. What is $|\SPS(1,\ldots,n-1,2^n)|$? 


\newpage

\item
(40 points) 
Recall that the AM-GM inequality is

For all $n\ge 2$, for all $x_1,\ldots,x_n\in\R^+$

$$\frac{x_1+\cdots+x_n}{n} \ge (x_1\cdots x_n)^{1/n}.$$

with equality iff $x_1=\cdots=x_n$. 

We are wondering: When are the Arithmetic Mean and the Geometric Mean
the furthest apart?

\begin{enumerate}
\item 
Write a program that will do the following:
\begin{enumerate}
\item
Input $n,N$
\item
For all subsets $\{x_1<\cdots<x_n\}$ of  $\{1,\ldots,N\}$ of size $n$ compute
$A=\frac{x_1+\cdots+x_n}{n}$, $G=(x_1\cdots x_n)^{1/n}$, $D=A-G$
\item
Then print out the top $n$ values of $D$ and which $x_1<\cdots<x_n$
lead to them.
\item \textbf{Send Code to Emily: ekaplitz@umd.edu}
\end{enumerate}
\item
(30 points) 
Run the program for $(n,N)=(5,10)$ and present the results. 
\item
(10 points) 
Run the program for $1 \leq n \leq 20$ and $n \leq N \leq 20$. Make a conjecture about what types of $x_1<\cdots<x_n$ lead to
large values of $D$. 
\end{enumerate} 



\end{enumerate}

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