\documentclass[12pt,HW250,ifthen]{article}
\usepackage{comment}
\newcommand{\ob}[1]{\frac{#1}{374}}
\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{\Rpos}{{\sf R}^+}
\newcommand{\goes}{\rightarrow}
\usepackage{amsmath}
\usepackage{bm}

\begin{document}

\centerline{\bf Take-Home Final Problems,}
\centerline{\bf Morally due Mon May 10, 9:00AM, NO Dead Cat}

\bigskip

On  Monday May 17, 8:00PM-10:15PM is your final. 
It will be 2.25-hours and be 70 points.  Why 70 points?
Because THIS document has THREE problems that you will do ahead of time and have
plenty of time to do, which is worth 30 points.

This must be handed in typed and easy to read.

\begin{enumerate}
\item (10 points) 
All quantifiers are over $\N$. 
Let $Q(n)$ be the statement

$$(\exists d_1 \le \cdots \le d_n)\biggl [ 1 = \sum_{i=1}^n \frac{1}{d_i^2}\biggr ]$$

(Note that the $d_i$'s do  NOT need to be distinct.) 


\begin{enumerate}
\item
(1 points) 
Prove that $Q(5)$ is FALSE. 
\item
(1 points)
Prove $Q(6)$ is true by giving 6 numbers $d_1,\ldots,d_6$ such that $1=\sum_{i=1}^6 \frac{1}{d_i^2}$.

{\it Hint} Write a program to find the numbers for your. 

\item 
(1 points)
Prove $Q(7)$ is true by giving 7 numbers $d_1,\ldots,d_7$ such that $1=\sum_{i=1}^7 \frac{1}{d_i^2}$.

{\it Hint} Write a program to find the numbers for your. 


\item 
(1 points)
Prove $Q(8)$ is true by giving 8 numbers $d_1,\ldots,d_8$ such that $1=\sum_{i=1}^8 \frac{1}{d_i^2}$.

{\it Hint} Write a program to find the numbers for your. 


\item
(6 points)
Show that $(\forall n\ge 8)[Q(n)\goes Q(n+3)]$. 


\end{enumerate} 

Do this problem on the next page and, if needed, the page after that.


\centerline{\bf GOTO NEXT PAGE} 


\newpage

 


\item 
(10 points) 

\begin{enumerate}
\item
(0 points but you will need this later)
Show that 
$$1= \frac{1}{2} + \frac{1}{3}+\frac{1}{6}.$$

\item 
(0 points but you will need this later)
Show that 
$$\frac{1}{6} = \frac{1}{8} + \frac{1}{24}.$$


\item 
(5 points) 

Use the first and second parts of this problem to prove 

{\bf The Reciprocal Theorem} 

For all $n\ge 3$ there exists $d_1<\cdots<d_n$ such that $\sum_{i=1}^n \frac{1}{d_i}=1$. 

\item
(5 point)
Use your proof and a computer program to write 1 as a sum of 10 reciprocals. 
Give us those 10. 
(You MUST use your proof.)
\end{enumerate}


\centerline{\bf GOTO NEXT PAGE}

\newpage 


\item 
(10 points- 5 each)
Let $L_1$ and $L_2$ be two linear orderings. 

An {\bf Order-Preserving bijection between $L_1$ and $L_2$} is 
a bijection $f$ with domain $L_1$, co-domain $L_2\Z$
such that $x<y \goes f(x) < f(y)$. 

For each of the following pairs of linear orderings say
if there is an order preserving bijection between them.

If YES then describe the order-preserving bijection.

If NO then prove there is no order-preserving bijection.


\begin{enumerate}
\item
$\N+\Z$ and $\Z+\N$. 
\item
$\Z+\Z+\Z$ and $\Z+\Z$. 
\end{enumerate}


\end{enumerate}
\end{document}