% 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}{1} % 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 4 of the 8 exercises!} \\
\textbf{Deadline:} October 9, 2024
\end{center}

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

\section{Exercise 1}

\subsection{Problem}

Let $\tup{f_0, f_1, f_2, \ldots}$ be the Fibonacci sequence
(which, as we recall, starts with $f_0 = 0$ and $f_1 = 1$
and continues by the recursive rule $f_n = f_{n-1} + f_{n-2}$
for all $n \geq 2$).
Furthermore, set $f_{-1} := 1$ and
$\ell_n := f_{n+1} + f_{n-1}$ for each $n \in \NN$.

\begin{enumerate}

\item[\textbf{(a)}] Prove that $\ell_0 = 2$ and
$\ell_1 = 1$ and $\ell_n = \ell_{n-1} + \ell_{n-2}$
for all $n \geq 2$.

\end{enumerate}

Let $m \in \NN$. Prove the following:

\begin{enumerate}

\item[\textbf{(b)}] We have
$f_{2m} = f_{m} \ell_{m}$.

\item[\textbf{(c)}] We have
$f_{2m+1} - 1 = f_{m} \ell_{m+1}$ if $m$ is even.

\item[\textbf{(d)}] We have
$f_{2m+1} - 1 = f_{m+1} \ell_{m}$ if $m$ is odd.

\end{enumerate}

\subsection{Solution}

...

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

\section{Exercise 2}

\subsection{Problem}

Define a sequence $\tup{a_0, a_1, a_2, \ldots}$ of
positive rational numbers recursively by setting
\begin{align*}
a_0 &= 1, \qquad a_1 = 1, \qquad a_2 = 1,
\qquad \text{and}\\
a_n &= \dfrac{a_{n-1} a_{n-2}}{a_{n-3} + 1}
\qquad \text{for each } n \geq 3.
\end{align*}
Prove that $a_n$ is the reciprocal of a positive
integer for each $n \in \NN$.
(For instance, $a_8 = \dfrac{1}{195840}$. Note the
subtle difference to Homework set \#0 Exercise 3.)

\subsection{Solution}

...

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

\section{Exercise 3}

\subsection{Problem}

For each $n \in \NN$, let $\ive{n}$ denote the set
$\set{1,2,\ldots,n}$.

If $P$ is a finite set of integers and if
$i \in \ive{\abs{P}}$, then $\min\nolimits_i P$ shall
denote the $i$-th smallest element of $P$. For
instance, $\min\nolimits_3\set{2,4,6,8,10,12} = 6$.

If $S$ and $T$ are two finite sets of integers, then
we say that $S \preccurlyeq T$ if and only if we have
\[
\abs{S} \geq \abs{T}
\qquad \text{and} \qquad
\tup{ \min\nolimits_i S \leq \min\nolimits_i T
\text{ for all } i \in \ive{\abs{T}} }.
\]
For instance, $\set{3,5,8,10} \preccurlyeq \set{4,5,9}$.

Let $S$ and $T$ be two finite sets of positive
integers. Prove the following:

\begin{enumerate}

\item[\textbf{(a)}] We have $S \preccurlyeq T$
if and only if each positive integer $k$ satisfies
$\abs{S \cap \ive{k}} \geq \abs{T \cap \ive{k}}$.

\item[\textbf{(b)}] Let $p$ be a positive integer.
Assume that $S \preccurlyeq T$.
Show that $\ive{p} \setminus T \preccurlyeq
\ive{p} \setminus S$.

\end{enumerate}

\subsection{Solution}

...

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

\section{Exercise 4}

\subsection{Problem}

Let $a_1, a_2, \ldots, a_n$ be $n$ distinct positive
integers. Prove that
\[
a_1^2 + a_2^2 + \cdots + a_n^2
\geq \dfrac{2n+1}{3}\tup{a_1+a_2+\cdots+a_n}.
\]
(Note that this is an equality when $a_i=i$ for all $i$,
due to the formulas
\[
1+2+\cdots+n = \dfrac{n\tup{n+1}}{2}
\qquad \text{and} \qquad
1^2+2^2+\cdots+n^2 = \dfrac{n\tup{n+1}\tup{2n+1}}{6}.
\]
)

\subsection{Solution}

...

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

\section{Exercise 5}

\subsection{Problem}

Using Euler's famous formula
$\sum_{i=1}^{\infty} \dfrac{1}{i^2}
= \dfrac{\pi^2}{6}$ (the answer to
\href{https://en.wikipedia.org/wiki/Basel_problem}{the Basel problem}),
prove that
\[
\sum_{i=1}^{\infty} \dfrac{1}{i^2\tup{ i+1 }}
= \dfrac{\pi^2}{6} - 1
\qquad \text{and}\qquad
\sum_{i=2}^{\infty} \dfrac{1}{i^2\tup{ i-1 }}
= 2 - \dfrac{\pi^2}{6}.
\]

\subsection{Solution}

...

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

\section{Exercise 6}

\subsection{Problem}

Let $n \geq 3$ be an integer. Prove that the
number $1$ can be written as
\[
1 = \dfrac{1}{a_1} + \dfrac{1}{a_2} + \cdots + \dfrac{1}{a_n}
\]
for some $n$ distinct positive integers
$a_1, a_2, \ldots, a_n$. (For instance, for $n = 3$,
we can write $1 = \dfrac12 + \dfrac13 + \dfrac16$.)

\subsection{Solution}

...


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

\section{Exercise 7}

\subsection{Problem}

Let $n$ be a positive integer. Let
$a_1, a_2, \ldots, a_n$ be $n$ distinct positive
reals. Consider the $2^n$ numbers of the
form $\pm a_1 \pm a_2 \pm \cdots \pm a_n$.
(There are $n$ many $\pm$ signs here, and each one
can be either a $+$ or a $-$ independently of the
others; thus, there are $2^n$ possible choices.)

Prove that at least $\dbinom{n+1}2$ of these
$2^n$ numbers are distinct.

\subsection{Solution}

...


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

\section{Exercise 8}

\subsection{Problem}

Let $\tup{x_1, x_2, x_3, \ldots }$ be a sequence
of integers that satisfies
\begin{align*}
x_1 &= 1 \qquad \text{and}\\
x_{2k} &= - x_k \qquad \text{for all } k \geq 1,
\qquad \text{and}\\
x_{2k-1} &= \tup{-1}^{k+1} x_k \qquad
\text{for all } k \geq 1.
\end{align*}

\begin{enumerate}

\item[\textbf{(a)}] Prove that
$x_1 + x_2 + \cdots + x_n
= x_{n+1} + x_{n+2} + \cdots + x_{4n}$
for each $n \in \NN$.

\item[\textbf{(b)}] Prove that
$x_1 + x_2 + \cdots + x_n \geq 0$ for each
$n \in \NN$.

\end{enumerate}

\subsection{Solution}

...



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

\begin{thebibliography}{99999999}                                                                                         %

\end{thebibliography}

\end{document}

