\documentclass[11pt]{amsart}
\usepackage{geometry}                % See geometry.pdf to learn the layout options. There are lots.
\geometry{letterpaper}                   % ... or a4paper or a5paper or ... 
\usepackage{amsfonts, amscd, amssymb, amsthm, amsmath, multicol, enumerate}
\usepackage{pdfsync}%leaves makers for tex searching

%%%%% Margins %%%%% 
\setlength{\topmargin}{1in} %top/bot margins
\setlength{\oddsidemargin}{\topmargin} %sidemargins

\setlength{\textheight}{11in} \setlength{\textwidth}{8.5in}
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%%% Environments %%%--------------------------------------------------------- 
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\newenvironment{summary}{\medskip \paragraph*{\emph{Summary}.}}{  \hfill \break  \rule{1.5cm}{0.4pt} \medskip }
\newcommand\Ans[1]{\hfill \emph{Answer: #1}}

%%% Pictures %%%--------------------------------------------------------- 
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\usepackage[pdftex,bookmarks]{hyperref}
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\newcommand\TikZ[1]{\begin{matrix}\begin{tikzpicture}#1\end{tikzpicture}\end{matrix}}



%%% Color  %%%--------------------------------------------------------------
\usepackage{color}
\definecolor{dred}{rgb}{.65, 0, 0.15}
\newcommand{\NOTE}[1]{{\color{blue}#1}}

%%% Alphabets %%%--------------------------------------------------------
\def\cA{\mathcal{A}}\def\cB{\mathcal{B}}\def\cC{\mathcal{C}}\def\cD{\mathcal{D}}\def\cE{\mathcal{E}}\def\cF{\mathcal{F}}\def\cG{\mathcal{G}}\def\cH{\mathcal{H}}\def\cI{\mathcal{I}}\def\cJ{\mathcal{J}}\def\cK{\mathcal{K}}\def\cL{\mathcal{L}}\def\cM{\mathcal{M}}\def\cN{\mathcal{N}}\def\cO{\mathcal{O}}\def\cP{\mathcal{P}}\def\cQ{\mathcal{Q}}\def\cR{\mathcal{R}}\def\cS{\mathcal{S}}\def\cT{\mathcal{T}}\def\cU{\mathcal{U}}\def\cV{\mathcal{V}}\def\cW{\mathcal{W}}\def\cX{\mathcal{X}}\def\cY{\mathcal{Y}}\def\cZ{\mathcal{Z}}

\def\AA{\mathbb{A}} \def\BB{\mathbb{B}} \def\CC{\mathbb{C}} \def\DD{\mathbb{D}} \def\EE{\mathbb{E}} \def\FF{\mathbb{F}} \def\GG{\mathbb{G}} \def\HH{\mathbb{H}} \def\II{\mathbb{I}} \def\JJ{\mathbb{J}} \def\KK{\mathbb{K}} \def\LL{\mathbb{L}} \def\MM{\mathbb{M}} \def\NN{\mathbb{N}} \def\OO{\mathbb{O}} \def\PP{\mathbb{P}} \def\QQ{\mathbb{Q}} \def\RR{\mathbb{R}} \def\SS{\mathbb{S}} \def\TT{\mathbb{T}} \def\UU{\mathbb{U}} \def\VV{\mathbb{V}} \def\WW{\mathbb{W}} \def\XX{\mathbb{X}} \def\YY{\mathbb{Y}} \def\ZZ{\mathbb{Z}}  

\def\ii{\mathbf{i}}\def\jj{\mathbf{j}}\def\kk{\mathbf{k}}\def\uu{\mathbf{u}}\def\vv{\mathbf{v}}\def\ww{\mathbf{w}}

\def\<{\langle} \def\>{\rangle}
\def\half{\hbox{$\frac12$}}
\def\id{\mathrm{id}}
\def\img{\mathrm{img}}
\def\vep{\varepsilon}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\HW{3}
\def\DUE{9/20/2018}
\def\NAME{YOUR NAME HERE}
\title[Homework \HW]{Homework \HW \\
Math 308\\
\small Due: \DUE}
\author{}
%\date{}                                           % Activate to display a given date or no date

\begin{document}
\maketitle %Comment out this line, and uncomment the following for your assignment:
%\begin{flushright}
%\NAME\\
%Math 308\\
%Homework \HW\\
%\DUE
%\end{flushright}


\begin{enumerate}[1.]
\item For each of (a)--(f), do the following:
\begin{enumerate}[(i)]
\item In the problem statement, what are the assumptions? What are the conclusions?
\item Do a couple of small examples, or do a similar smaller/easier version of the problem.
\item Solve the problem. As always, your writeup should use the writing skills you developed in Chapters 3--4. \\
{[Hint: Any time you see a root (square, seventh, whatever), try to get rid of it.]}
\end{enumerate}

\begin{enumerate}[(a)]
\item Show that 
$$\frac{a + b}{2} \geq \sqrt{ab} \qquad \text{ for all } \quad 0 < a \leq b.$$
\item Show that 
$$a^2 + b^2 + c^2 \geq ab + bc + ca$$
for all positive integers $a$, $b$, and $c$. 
\item Let $f(x) = 1/(1-x)$. Define the function $f^r$ by
$$f^r(x) := \underbrace{f(f(\cdots(f(f}_{r \text{ times}}(x)))\cdots)).$$
Find $f^{653}(56)$. 
\item Without using a calculator, show that $\sqrt[7]{7!} < \sqrt[8]{8!}$. \\
\item Without using a calculator, show that $\sqrt{100001} - \sqrt{100000} < \frac{1}{2 \sqrt{100000}}$.\\
\item Bottle $A$ contains a liter of milk and bottle $B$ contains a liter of coffee. A spoonful of coffee from $B$ is poured into $A$, and the contents are mixed well. Liquid from $A$ is then poured into $B$ until $B$ has one liter of liquid. Is the fraction of coffee in $A$ greater than the fraction of milk in $B$, or vice versa? \\
{[Hint: Giving things names will be super important here! You may also assume that your spoon isn't giant.]}
\end{enumerate}




\end{enumerate}


\thispagestyle{empty}
\end{document}