% LaTeX solution template for Math 235
% -----------------------------------------------------------------

% Like most advanced LaTeX files, this one begins with a lot of
% boilerplate. You don't need to understand (or even read) most of it.
% All you need to do is fill in your name, email address,
% and the number of the pset. (Search for "METADATA" to find the place
% for this.) Then, you can go straight to the "EXERCISE 1"
% section and start writing your solutions.
% The "VARIOUS USEFUL COMMANDS" section is probably worth taking a
% look at at some point.

% If you are here to learn about LaTeX, search for
% "START READING HERE" and start reading there.

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%	PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
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\documentclass[paper=a4, fontsize=12pt]{scrartcl} % A4 paper and 12pt font size
\usepackage[T1]{fontenc} % Use 8-bit encoding that has 256 glyphs
\usepackage[english]{babel} % English language/hyphenation
\usepackage{amsmath,amsfonts,amsthm,amssymb} % Math packages
\usepackage{mathrsfs}    % More math packages
\usepackage{sectsty}  % Allows customizing section commands
\allsectionsfont{\centering \normalfont\scshape} % Make all section titles centered, the default font and small caps %remove this to left align section tites
\usepackage{hyperref} % Turns cross-references into hyperlinks,
                      % and defines \url and \href commands.
\usepackage{graphicx} % For embedding graphics files.
\usepackage{framed}   % For the "leftbar" environment used below.
\usepackage{ifthen}   % Used for the \powset command below.
\usepackage{lastpage} % for counting the number of pages
\usepackage[headsepline,footsepline,manualmark]{scrlayer-scrpage}
\usepackage[height=10in,a4paper,hmargin={1in,0.8in}]{geometry}
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{tikz}     % This is a powerful tool to draw vector
                      % graphics inside LaTeX. In particular, you can
                      % use it to draw graphs.
\usepackage{verbatim} % For the "verbatim" environment, in which
                      % special symbols can be used freely without
                      % confusing the compiler. (And it's typeset in
                      % a constant-width font.)
                      % Useful, e.g., for quoting code (or ASCII art).

%\numberwithin{table}{section} % Number tables within sections (i.e. 1.1, 1.2, 2.1, 2.2 instead of 1, 2, 3, 4)

\setlength\parindent{20pt} % Makes indentation for paragraphs longer.
                           % This makes paragraphs stand out more.

%----------------------------------------------------------------------------------------
%	VARIOUS USEFUL COMMANDS
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% The commands below might be convenient. For example, you probably
% prefer to write $\powset[2]{V}$ for the set of $2$-element subsets
% of $V$, rather than writing $\mathcal{P}_2(V)$.
% Notice that you can easily define your own commands like this.
% Caveat: Some of these commands need to be properly "guarded" when
% they occur in subscripts or superscripts. So you should not write
% $K_\CC$, but rather $K_{\CC}$.
\newcommand{\CC}{\mathbb{C}} % complex numbers
\newcommand{\RR}{\mathbb{R}} % real numbers
\newcommand{\QQ}{\mathbb{Q}} % rational numbers
\newcommand{\NN}{\mathbb{N}} % nonnegative integers
\newcommand{\Z}[1]{\mathbb{Z}/#1\mathbb{Z}} % integers modulo k
                                            % (syntax: "\Z{k}")
\newcommand{\ZZ}{\mathbb{Z}} % integers
\newcommand{\id}{\operatorname{id}} % identity map
\newcommand{\lcm}{\operatorname{lcm}}
% Lowest common multiple. For historical reasons, LaTeX has a \gcd
% command built in, but not an \lcm command. The preceding line
% rectifies that.
\newcommand{\rev}{\operatorname{rev}} % reversal of a walk
\newcommand{\powset}[2][]{\ifthenelse{\equal{#2}{}}{\mathcal{P}\left(#1\right)}{\mathcal{P}_{#1}\left(#2\right)}}
% $\powset[k]{S}$ stands for the set of all $k$-element subsets of
% $S$. The argument $k$ is optional, and if not provided, the result
% is the whole powerset of $S$.
\newcommand{\set}[1]{\left\{ #1 \right\}}
% $\set{...}$ compiles to {...} (set-brackets).
\newcommand{\abs}[1]{\left| #1 \right|}
% $\abs{...}$ compiles to |...| (absolute value, or size of a set).
\newcommand{\tup}[1]{\left( #1 \right)}
% $\tup{...}$ compiles to (...) (parentheses, or tuple-brackets).
\newcommand{\ive}[1]{\left[ #1 \right]}
% $\ive{...}$ compiles to [...] (Iverson bracket, aka truth value).
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
% $\floor{...}$ compiles to |_..._| (floor function).
\newcommand{\verts}[1]{\operatorname{V}\left( #1 \right)}
% $\verts{...}$ compiles to V(...) (vertex set of a graph/digraph).
\newcommand{\edges}[1]{\operatorname{E}\left( #1 \right)}
% $\edges{...}$ compiles to E(...) (edge set of a graph).
\newcommand{\arcs}[1]{\operatorname{A}\left( #1 \right)}
% $\arcs{...}$ compiles to A(...) (arc set of a digraph).
\newcommand{\lf}[2]{#1^{\underline{#2}}}
% $\lf{...1}{...2}$ compiles to $...1^{\underline{...2}}$.
% This is a notation for the falling factorial.
\newcommand{\underbrack}[2]{\underbrace{#1}_{\substack{#2}}}
% $\underbrack{...1}{...2}$ yields
% $\underbrace{...1}_{\substack{...2}}$. This is useful for doing
% local rewriting transformations on mathematical expressions with
% justifications. For example, try this out:
% $ \underbrack{(a+b)^2}{= a^2 + 2ab + b^2 \\ \text{(by the binomial formula)}} $
\newcommand{\horrule}[1]{\rule{\linewidth}{#1}} % Create horizontal rule command with 1 argument of height
\newcommand{\nnn}{\nonumber\\} % Don't number this line in an "align" environment, and move on to the next line.
\newcommand{\explain}[1]{\tup{\hspace{-0.5pc}\text{\begin{tabular}{c}#1\end{tabular}\hspace{-0.5pc}}}}
% $\explain{...}$ interprets the "..." as text (you can
% nest some math inside by putting it in $...$'s as you
% would in any other text block) and puts it in parentheses.

%----------------------------------------------------------------------------------------
%	MAKING SUMMATION SIGNS ALWAYS PUT THEIR BOUNDS ABOVE AND BELOW
%	THE SIGN
%----------------------------------------------------------------------------------------
% The following are hacks to ensure that sums (such as
% $\sum_{k=1}^n k$) always put their bounds (i.e., the $k=1$ and the
% $n$) underneath and above the sign, as opposed to on its right.
% Same for products (\prod), set unions (\bigcup) and set
% intersections (\bigcap). Remove the 8 lines below if you do not want
% this behavior.
\let\sumnonlimits\sum
\let\prodnonlimits\prod
\let\cupnonlimits\bigcup
\let\capnonlimits\bigcap
\renewcommand{\sum}{\sumnonlimits\limits}
\renewcommand{\prod}{\prodnonlimits\limits}
\renewcommand{\bigcup}{\cupnonlimits\limits}
\renewcommand{\bigcap}{\capnonlimits\limits}

%----------------------------------------------------------------------------------------
%	ENVIRONMENTS
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% The incantations below define how theorem environments
% (\begin{theorem} ... \end{theorem}) and their likes will look like.
\newtheoremstyle{plainsl}% <name>
  {8pt plus 2pt minus 4pt}% <Space above>
  {8pt plus 2pt minus 4pt}% <Space below>
  {\slshape}% <Body font>
  {0pt}% <Indent amount>
  {\bfseries}% <Theorem head font>
  {.}% <Punctuation after theorem head>
  {5pt plus 1pt minus 1pt}% <Space after theorem headi>
  {}% <Theorem head spec (can be left empty, meaning `normal')>

% Environments which make the text inside them slanted:
\theoremstyle{plainsl}
  \newtheorem{theorem}{Theorem}[section]
  \newtheorem{proposition}[theorem]{Proposition}
  \newtheorem{lemma}[theorem]{Lemma}
  \newtheorem{corollary}[theorem]{Corollary}
  \newtheorem{conjecture}[theorem]{Conjecture}
% Environments that don't:
\theoremstyle{definition}
  \newtheorem{definition}[theorem]{Definition}
  \newtheorem{example}[theorem]{Example}
  \newtheorem{exercise}[theorem]{Exercise}
  \newtheorem{examples}[theorem]{Examples}
  \newtheorem{algorithm}[theorem]{Algorithm}
  \newtheorem{question}[theorem]{Question}
 \theoremstyle{remark}
  \newtheorem{remark}[theorem]{Remark}
\newenvironment{statement}{\begin{quote}}{\end{quote}}
\newenvironment{fineprint}{\begin{small}}{\end{small}}

%----------------------------------------------------------------------------------------
%	METADATA
%----------------------------------------------------------------------------------------
\newcommand{\myname}{Darij Grinberg} % ENTER YOUR NAME HERE
\newcommand{\mymail}{darij.grinberg@drexel.edu} % ENTER YOUR EMAIL HERE
\newcommand{\psetnumber}{6} % ENTER THE NUMBER OF THIS PSET HERE

%----------------------------------------------------------------------------------------
%	HEADER AND FOOTER
%----------------------------------------------------------------------------------------
\ihead{Solutions to homework set \#\psetnumber} % Page header left
\ohead{page \thepage\ of \pageref{LastPage}} % Page header right
\ifoot{\myname} % left footer
\ofoot{\mymail} % right footer

%----------------------------------------------------------------------------------------
%	TITLE SECTION
%----------------------------------------------------------------------------------------
\title{	
\normalfont \normalsize 
\textsc{Drexel University, Department of Mathematics} \\ [25pt] % Your university, school and/or department name(s)
\horrule{0.5pt} \\[0.4cm] % Thin top horizontal rule
\huge Math 235: Mathematical Problem Solving, \\
Fall 2024:
Homework \psetnumber\\% The assignment title
\horrule{2pt} \\[0.5cm] % Thick bottom horizontal rule
}
\author{\myname}

\begin{document}

\maketitle % This command causes the title part to be printed.

\begin{center}
\Large
\textbf{Please solve 5 of the 10 exercises!} \\
\textbf{Deadline:} November 20, 2024
\end{center}

%----------------------------------------------------------------------------------------
%	EXERCISE 1
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 1}

\subsection{Problem}

Let $\tup{a_0, a_1, a_2, \ldots}$ be a sequence of
positive integers such that
\[
a_n = \tup{a_{n-1}^2 \% a_{n-2}} + 1
\qquad \text{for each } n \geq 2.
\]
Prove that this sequence is eventually $2$-periodic
(i.e., there exists some $m \in \NN$ such that the
subsequence $\tup{a_m, a_{m+1}, a_{m+2}, \ldots}$
is $2$-periodic).

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 2
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 2}

\subsection{Problem}

Find all pairs $\tup{x,y}$ of nonnegative integers
such that $\abs{x^2 - xy - y^2} = 5$.
(Recall the Lucas sequence from
Example 4.9.3 in
\href{https://www.cip.ifi.lmu.de/~grinberg/t/20f/mps.pdf}{the notes}.)

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 3
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 3}

\subsection{Problem}

Let $n$ be a positive integer. Let $a$ be any
integer. Prove that there exist $i \in \NN$ and
$x \in \ZZ$ such that $a^{i+1} x \equiv a^i \mod n$.

\subsection{Remark}

This $x$ can be viewed as a weak version of a modular
inverse of $a$ modulo $n$. An actual modular inverse
would satisfy the congruence $ax \equiv 1 \mod n$
(that is, $a^{i+1} x \equiv a^i \mod n$ for $i=0$),
but it only exists when $a \perp n$ (see
Theorem 3.5.9 in
\href{https://www.cip.ifi.lmu.de/~grinberg/t/20f/mps.pdf}{the notes}),
whereas the
weak version always exists according to this exercise.

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 4
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 4}

\subsection{Problem}

Let $n$ be a positive integer. Let
$x_1, x_2, \ldots, x_{n+2}$ be $n+2$ integers.
Prove that there exist two distinct elements $i$
and $j$ of $\set{1, 2, \ldots, n+2}$ such that
$x_i - x_j$ or $x_i + x_j$ (or both) is
divisible by $2n$.

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 5
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 5}

\subsection{Problem}

Inside a regular hexagon with sidelength $1$,
you have marked $7$ points. Show that there are
two marked points whose distance to each other
is at most $1$.

\subsection{Solution}

...

%----------------------------------------------------------------------------------------
%	EXERCISE 6
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 6}

\subsection{Problem}

Let $a \in \ZZ$ be such that $a \equiv 1 \mod 3$.
Let $k$ be a positive integer.

\begin{enumerate}

\item[\textbf{(a)}] Prove that there exists a
unique $x \in \set{0, 1, \ldots, 3^k-1}$ such that
$x \equiv 1 \mod 3$ and $x^2 \equiv a \mod 3^k$.

\item[\textbf{(b)}] Prove that for each
$i \in \NN$, there exists a unique
$x_i \in \set{0, 1, \ldots, 3^k-1}$ such that
$x_i \equiv 1 \mod 3$ and
$x_i^{2^i} \equiv a \mod 3^k$.

\item[\textbf{(c)}] Consider the sequence
$\tup{x_0, x_1, x_2, \ldots}$ of these unique
$x_i$'s. Is this sequence periodic?

\end{enumerate}

\subsection{Remark}

The $x$ in part \textbf{(a)} can be regarded as the
``canonical''\ square root of $a$ modulo $3^k$.
In general, square roots modulo $n$ neither
necessarily exist nor are always unique, just like
square roots of real numbers; thus the claim of part
\textbf{(a)} is rather remarkable.

\subsection{Solution}

...


%----------------------------------------------------------------------------------------
%	EXERCISE 7
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 7}

\subsection{Problem}

Let $a_1, a_2, \ldots, a_m$ be $m$ integers, and
let $n$ be a positive integer such that $n < 2^m$.
Prove that we can find $m$ numbers
$x_1, x_2, \ldots, x_m \in \set{0, 1, -1}$ such
that not all these numbers
$x_1, x_2, \ldots, x_m$ are zero and such that
$n \mid \sum_{i=1}^m x_i a_i$. (In other words,
prove that we can make the sum
$a_1 + a_2 + \cdots + a_m$ divisible by $n$ if we
are allowed to throw some (not all) of the
addends away and replace some (possibly none,
possibly all) of the remaining addends by their
negatives.) \medskip

[\textbf{Example:} Let $m=4$ and $n=13$ and
$\tup{a_1, a_2, \ldots, a_m} = \tup{4,7,19,40}$.
Then, the four numbers $0,-1,1,1$ are $m$ numbers
$x_1, x_2, \ldots, x_m$ that satisfy
$n \mid \sum_{i=1}^m x_i a_i$, since
$13 \mid 0 \cdot 4 + \tup{-1} \cdot 7 + 1 \cdot
19 + 1 \cdot 40$.]

\subsection{Solution}

...


%----------------------------------------------------------------------------------------
%	EXERCISE 8
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 8}

\subsection{Problem}

Let $a_1, a_2, \ldots, a_k$ be $k$ real numbers.
Let $\tup{x_0, x_1, x_2, \ldots}$ be a sequence
of real numbers that is
$\tup{a_1, a_2, \ldots, a_k}$-recurrent (see
Definition 4.9.24 in
\href{https://www.cip.ifi.lmu.de/~grinberg/t/20f/mps.pdf}{the notes}).
Assume that this sequence has only finitely many
distinct entries (i.e., the set
$\set{x_0, x_1, x_2, \ldots}$ is finite). Show
the following:

\begin{enumerate}

\item[\textbf{(a)}] The sequence
$\tup{x_0, x_1, x_2, \ldots}$ is eventually
periodic (i.e., there exists some $m \in \NN$
such that the sequence
$\tup{x_m, x_{m+1}, x_{m+2}, \ldots}$ is
periodic).

\item[\textbf{(b)}] If $a_k \neq 0$, then the
sequence $\tup{x_0, x_1, x_2, \ldots}$
is periodic.

\end{enumerate}

\subsection{Solution}

...


%----------------------------------------------------------------------------------------
%	EXERCISE 9
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 9}

\subsection{Problem}

Let $n > 1$. Let $A$ be an $n \times n$-matrix whose
entries are the $n^2$ integers $1, 2, \ldots, n^2$
in some arbitrary order.

\begin{enumerate}

\item[\textbf{(a)}] Prove that we can find two
entries $i$ and $j$ of $A$ that lie in the same row
and satisfy $\dfrac{n-1}{n} \leq \dfrac{i}{j} < 1$.

\item[\textbf{(b)}] Prove that we can find two
entries $i$ and $j$ of $A$ that lie in the same row
or the same column and satisfy
$\dfrac{n}{n+1} \leq \dfrac{i}{j} < 1$.

\end{enumerate}

\subsection{Solution}

...


%----------------------------------------------------------------------------------------
%	EXERCISE 10
%----------------------------------------------------------------------------------------
\horrule{0.3pt} \\[0.4cm]

\section{Exercise 10}

\subsection{Problem}

Let $a$ and $n$ be two integers such that $n > 0$.
Let $u$ and $v$ be two positive integers such that
$uv > n$ and $v > 1$. Prove that there exist two
integers $x$ and $y$ with $\abs{x} < u$ and
$0 < y < v$ and $ay \equiv x \mod n$.

\subsection{Solution}

...



\horrule{0.3pt} \\[0.4cm]

\begin{thebibliography}{99999999}                                                                                         %
\end{thebibliography}

\end{document}

