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\noindent \textbf{Math 315 - Fall 2017 \\
Homework 1} \\
Due September 11, 2017 \vspace{-4mm}\\
\noindent \small \textit{There is no problem in all mathematics that cannot be solved by direct counting.}\\\vspace{1mm} \hfill---
Ernst Mach
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\noindent \textbf{Turn in:}
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\begin{enumerate}
\item Show that if $n+1$ distinct integers are chosen from the set $\{1, 2, \dots, 2n\}$, then there are
always two whose greatest common divisor is 1.
\bigskip
\item (Exercise 24, Chapter 1) Find all 4-tuples $(a, b, c, d)$ of distinct positive integers so that $a < b < c < d$ and
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = 1\]
{\em Hint:} Look at the solution to Exercise 2 in Chapter 1.
\bigskip
\item Let $S$ be a set of 17 points inside a cube of side length 1. Prove that there exists a sphere of radius 1/2 which
encloses at least three of the points.
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\item For each $n$ describe a sequence of $n^2$ numbers which does not contain a monotone increasing sequence of length $n+1$, nor a monotone decreasing sequence of length $n+1$. This shows that the Erd\H{o}s-Szekeres Theorem is tight.
\end{enumerate}
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