\documentclass{amsart} % We are using the AMS class for hte extra
                       % mathematical bits
\usepackage{amsthm}
\usepackage{amssymb}

\newtheorem{thm}{Theorem}
  \theoremstyle{definition}
  \newtheorem{problem}[thm]{Problem}


\begin{document}

\title{Math 300, In class problems\\
Week 1}
\author{Marius Ionescu}
\maketitle
\begin{problem}
If $(a_{n})$ is a sequence of real numbers, then in any closed interval
of $\mathbb{R}$ with positive length there exists a real number $x$
such that $x\ne a_{n}$ for each $n$.
\end{problem}

\begin{problem}
Let $f$ be a strictly increasing mapping of $\mathbb{R}$ onto $\mathbb{R}^{+}$
such that $f(0)=1$ and $f(x+y)=f(x)f(y)$. Prove that $f(x)=a^{x}$,
where $a=f(1)>1$.
\end{problem}

\begin{problem}
Let $(a_{n})$ be a sequence in $\mathbb{R}$. Prove that if the
three subsequences $(a_{2n})$, $(a_{2n+1})$, and $(a_{3n})$ are
convergent, then so is $(a_{n})$.
\end{problem}

\begin{problem}
What is the last digit of $3^{4798}$?
\end{problem}

\begin{problem}
You have a well-shuffled deck of 52 playing cards. You ask a friend
to divide the deck into 3 piles. You then be that one of the top $3$
cards is a face card (Jack, Queen, or King). Who has a better chance
of winning, you or your friend?
\end{problem}

\begin{problem}
Let $f$ be a real-valued function on the plane such that for every
square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow
that $f(P)=0$ for all points $P$ in the plane?
\end{problem}

\end{document}
