\documentclass{article}
\usepackage{amsmath}
\begin{document}
{\bf Worksheet \#3; date: 09/05/2018}
{\bf MATH 55 Discrete Mathematics}
\begin{enumerate}
\item {\em (Rosen 1.7.7)} Use a direct proof to show that every odd integer is the difference of two squares.
\item {\em (Rosen 1.7.8)} Prove that if $n$ is a perfect square, then $n+2$ is not a perfect square.
\item {\em (Rosen 1.7.13)} Prove that if $x$ is irrational, then $1/x$ is irrational.
\item {\em (Rosen 1.7.39)} Prove that at least one of the real numbers $a_1, a_2, \ldots, a_n$ is greater than or equal to the average of these numbers. What kind of proof did you use?
\item {\em (Rosen 1.8.17)} Suppose that $a$ and $b$ are odd integers with $a \ne b$. Show there is a unique integer $c$ such that $|a - c| = |b - c|$.
\item {\em (Rosen 1.8.25)} Write the numbers $1, 2, \ldots, 2n$ on a blackboard, where $n$ is an odd integer. Pick any two of the numbers, $j$ and $k$, write $|j-k|$ on the board and erase $j$ and $k$. Continue this process until only one integer is written on the board. Prove that this integer must be odd.
\item {\em (Rosen 1.8.34)} Prove that $\sqrt[3]{2}$ is irrational.
\item {\em (Rosen 1.8.43)} Prove that you can use dominoes to tile a rectangular checkerboard with an even number of squares.
\end{enumerate}
\end{document}