\documentclass[11pt]{article}
\usepackage{fullpage}
\usepackage{amssymb,amsfonts,amsmath,amsthm} 
\usepackage[margin= 1in]{geometry}
\usepackage{url}

\usepackage{enumerate}
\usepackage{color}
\usepackage[pdftex]{graphicx}
\usepackage{lastpage}
\usepackage{mathptmx}

\newtheorem{thm}{THEOREM}[]
\newtheorem{lem}[thm]{LEMMA}
\newtheorem{cor}[thm]{COROLLARY}
\newtheorem{prop}[thm]{PROPOSITION}
\newtheorem{as}[thm]{ASSUMPTION}
\newtheorem{conjecture}[thm]{Conjecture}
\newtheorem{claim}[thm]{CLAIM}
\theoremstyle{definition}
\newtheorem{defi}[]{DEFINITION}
\theoremstyle{definition}
\newtheorem{remark}[thm]{REMARK}
\newtheorem{fact}[thm]{Fact}
\newtheorem{ex}[thm]{Example}

\setlength{\parindent}{0cm}

\newcommand{\tab}{\hspace{3ex}}
\newcommand{\R}{\mathbb{R}}

\usepackage{titlesec}
\usepackage{sectsty}
\hyphenchar\font=-1

\sectionfont{  \normalsize	\sectionrule{0pt}{0pt}{-5pt}{0.05pt}
	}

\titleformat*{\section}{ \Large  \center}
\titleformat{\subsection}[runin]
               {\normalfont\bfseries }
               {}{0em}{}[.]
               
\titlespacing{\section}
               {\parindent}{0pt}{5pt}
\titlespacing{\subsection}
               {0pt}{2ex plus .1ex minus .2ex}{5pt} 
               
               
               \makeatletter
 \renewcommand{\@evenfoot}{%
   Page \thepage{} of \pageref{VeryLastPage} \hfil
    
 }
 \renewcommand{\@oddfoot}{\@evenfoot}
 \makeatother

               
\begin{document}

\begin{center}
{\huge  \sc   Homework 3} \\
\smallskip
{ Due Date: 04/19/2022}\\
\smallskip
{Name:  }\\
{\large  }
\end{center}
\vspace{1cm}



\begin{enumerate}
\item[8.1]
Prove by induction that it is possible to pay, without requiring change, any whole number of roubles greater than 7 with banknotes of value 3 roubles and 5 roubles.
\newpage 
\item[8.2] Prove by induction that 
\[
\sum_{r=1}^nr^2=\frac{1}{6}n(n+1)(2n+1).
\]
Deduce formulae for 
\[
1\cdot 1+2\cdot 3+\ldots +n(2n-1)
\]
and
\[
1^2+3^2+5^2+\ldots +(2n-1)^2.
\]
\newpage
\item[8.3] 
\begin{enumerate}
\item Work out 1, 1 + 8, 1 + 8 + 27 and 1 + 8 + 27 + 64. Guess a formula

$\sum_{r=1}^nr^3 $
and prove it.

\newpage
\item Check that 1 = 0+1, 2+3+4 = 1+8 and 5+6+$\ldots$+9 = 8+27.
Find a general formula for which these are the first three cases. Prove
your formula is correct.

\end{enumerate}
\newpage
\item[8.5] Prove the following statements by induction:

\begin{enumerate}
\item For all integers $n \geq 0$, the number $5^{2 n}-3^{n}$ is a multiple of 11 .
\item For any integer $n \geq 1$, the integer $2^{4 n-1}$ ends with an 8 .
\item The sum of the cubes of three consecutive positive integers is always a multiple of 9 .
\item If $x \geq 2$ is a real number and $n \geq 1$ is an integer, then $x^{n} \geq n x$.
\item If $n \geq 3$ is an integer, then $5^{n}>4^{n}+3^{n}+2^{n}$.
\end{enumerate}
\newpage

\item[8.9] 


Prove that if $0 < q <\frac{1}{2}$ then for all $n \geq 1$,
\[
(1+q)^n \leq 1+2^nq
\]

\newpage

\item[8.11] Just for this question, count 1 as a prime number. A well-known result in
number theory says that for every integer $x \geq 3$, there is a prime number
$p$ such that $\frac{x}{2} < p < x$. Using this result and strong induction, prove
that every positive integer is equal to a sum of primes, all of which are
different.


\newpage

\item[8.13]

\begin{enumerate}

\item  Suppose we have $n$ straight lines in a plane, and all the lines pass
through a single point. Into how many regions do the lines divide the
plane? Prove your answer.
\newpage 
\item  We know from Example 8.8 that n straight lines in general position
in a plane divide the plane into $\frac{1}{2}(n^2+n+2)$ regions. How many of
these regions are infinite and how many are finite?
(In case of any confusion, a finite region is one that has finite area; an
infinite region is one that does not.)

\end{enumerate}
\end{enumerate}




\label{VeryLastPage}
\end{document}