\documentclass{amsart} % we are using the AMS (American Mathematical
                       % Society) class, because it incorporates many
                       % extra mathematical bits
\usepackage{amsthm} % AMS theorem environments
\usepackage{amstext} %AMS text and mathematical symbols
\usepackage{amssymb}


%% Define the environment "Problem" based on the environment Theorem
\newtheorem{thm}{Theorem}
  \theoremstyle{definition}
  \newtheorem{problem}[thm]{Problem}



\begin{document}

\title{Math 399, Homework \#1}

\date{Due January 24, 2011}
\author{Marius Ionescu}
\maketitle % we need \maketitle after the \title,\date, and \author

\begin{problem}
If $f(x)$ is differentiable at $x=a$ and $f(a)\neq0$, find \[
\lim_{n\to\infty}\left(\frac{f(a+\frac{1}{n})}{f(a)}\right)^{n}\ .\]
\end{problem}

\begin{problem}
Let $X=[0,1]$ and let $\Delta=\{(x,x)\in X\times X\}$ be the diagonal
of the unit square. Assume that $f:X\times X\setminus\Delta\to\mathbb{R}$
and $g:X\to\mathbb{R}$ are continuous, and that\begin{eqnarray*}
\lim_{y\to x}f(x,y) & = & g(x)\;\text{and}\\
\lim_{x\to y}f(x,y) & = & g(y).\end{eqnarray*}
Dos it follow that $f$ has a continuous extension to $X\times X$?
If yes, prove it. If no, provide an example for which the statement
fails.
\end{problem}

\begin{problem}
Prove that if $a>1$, then $\lim_{n\to\infty}\bigl(n^{-1}\log_{a}n\bigr)=0$.
\end{problem}

\begin{problem}
Evaluate \[
\sum_{n=0}^{\infty}\frac{1}{\binom{2n}{n}}.\]
Hint: Find an integral formula for $\binom{2n}{n}^{-1}$.
\end{problem}

\begin{problem}
Show that given a sequence of $n^{2}+1$ distinct integers, either
there is an increasing subsequence of $n+1$ terms or a decreasing
subsequence of $n+1$ terms.
\end{problem}

\begin{problem}
The points of the plane are colored either red or blue. Prove that
for some color there are pairs of points at every possible distance.
\end{problem}

\begin{problem}
Prove that there exist irrational numbers $a$ and $b$ such that
$a^{b}$ is rational.
\end{problem}

\begin{problem}
Let $P$ be any point inside an equilateral triangle. Prove that
the sum of the distances from $P$ to each side of the triangle is
equal to the height of the triangle.
\end{problem}

\end{document}
