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{\bf Worksheet \#14; date: 10/15/2018}
{\bf MATH 55 Discrete Mathematics}
\begin{enumerate}
\item {\em (Rosen 6.2.13a)} Show that if five integers are selected from the first eight positive integers, there must be a pair of these integers with a sum equal to $9$.
\item {\em (Rosen 6.2.23)} Show that whenever $25$ girls and $25$ boys are seated around a circular table there is always a person both of whose neighbors are boys.
\item {\em (Rosen 6.2.40)} Prove that at a party where there are at least two people, there are two people who know the same number of other people there.
\item {\em (Rosen 6.2.45; challenging)} Let $x$ be an irrational number. Show that for some positive integer $j$ not exceeding the positive integer $n$, the absolute value of the difference between $jx$ and the nearest integer to $jx$ is less than $1/n$.
\end{enumerate}
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