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\title{\sc Math 2100 -- Homework 4}
\author{
	Fall 2022\\[10pt]
	due Wednesday, \textbf{November 2}, at the start of class\\[10pt]
	Sections 2.3, 2.4, 2.5, some 3.1
}
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\begin{center} 
	\emph{This homework assignment was written in \LaTeX{}. You can find the source code on the course website.}
\end{center}

\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. Please write the questions in the correct order. Explain all reasoning.

\begin{enumerate}	
	\item Use induction to prove that for all integers $n \geq 0$, the quantity $2^{2n+1} + 5^{2n+1}$ is divisible by $7$.

	\item Prove that $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$.

	\item Prove that there exists a positive integer $n$ such that $\frac{1}{n\ln(n)} < 0.0001$.

	\item Prove that any real number $r$ that makes the equation $r - \ds\frac{1}{r}=5$ true must be irrational.

	\item Prove that at a completely full Milwaukee Bucks game at the Fiserv Forum, there \emph{must} be at least two people that have both the same birthday \emph{and} the same first initial. (Note: you will have to look up the capacity of the arena!)

	\item Prove that if $a + b + c \geq 35$, then either $a \geq 10$, $b \geq 12$, or $c \geq 13$.


	\item Use the pigeonhole principle to prove that given any five integers, there will be two that have a sum or difference divisible by 7.

	\item Prove that if any five points other than $(0,0)$ are placed on the coordinate plane, then there are two points, call them $A$ and $B$, such that the angle formed by the rays from $(0,0)$ to $A$ and from $(0,0)$ to $B$ is acute.
		
	
	\item \sout{Write each of the following sets in set-builder notation.}
	\begin{enumerate}
		\item \sout{The set $S$ of integers that are multiples of 3 and a prefect square.}
		\item \sout{The set $T$ of positive integers that are bigger than $10$ and whose ones digit is a 5.}
		\item \sout{The set $R$ of real numbers whose square is a rational number.}
	\end{enumerate}
	
	\item \sout{List five elements in each of the following sets, unless there are fewer than 5 elements in the set (in which case, justify how you know you've listed all of the elements).}
	\begin{enumerate}
		\item \sout{$A = \{x \in \R : x^2 \in \N\}$}
		\item \sout{$B = \{S \subseteq \{1,2,3,4\} : \text{the sum of the elements of $S$ is even}\}$}
		\item \sout{$C = \{q \in \N : q = 2k \text{ for some $k\in\N$ and }q=2\ell+1\text{ for some $\ell\in\N$}\}$}
	\end{enumerate}


	
\end{enumerate}
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