% This template was created by Stephen on 27th March 2000
%
% Title:  Euclidean Geometry
% Author: Stephen Farrar
% Date:   16/4/2000
%

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% A command to generate the header text
\newcommand{\myheader}
{\footnotesize April 2006}

% Change the headers and footers
\pagestyle{myheadings} \markboth{\myheader}{\myheader}

% New Commands to make numbered lists easier
\newcommand{\be}{\begin{enumerate}}
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%--------------------------------------------------------
% Let the document begin!
\begin{document}

\begin{centre}
  \bf \Large
  Diophantine Equations
\end{centre}

\begin{enumerate}

\item Suppose that $ab=cd$. Show that there exist integers
$w$,$x$,$y$ and $z$ such that $a=wx$, $b=yz$, $c=xy$, $d=zx$. This
technique can be generalised to equations such as $ab=pcd$ for $p$
prime. For this, there are two cases according to whether $p$
divides $a$ or $b$.

\item Find all positive integer solutions to the equation
$x^2(x^2+y)=y^{m+1}$.

\item (Engel) Do there exist integers $m$, $n$ with
$m^2+(m+1)^2=n^4+(n+1)^4$?

\item (Engel) Find all integers $k$ such that there exist positive
integer solutions to the equation $x^2+y^2+z^2=kxyz.$

\item Parametrise all three term arithmetic progressions of
squares.

\item (Engel) If $ab=cd\neq0$ then $a^2+b^2+c^2+d^2$ is composite.

\item (1999 IMO Shortlist) Prove that there exists two strictly
increasing sequences $(a_n)$ and $(b_n)$ such that $a_n(a_n+1)$
divides $b_n^2+1$ for every natural number $n$.

\item (1999 Taiwan) Determine all triples $(x,y,z)$ of positive
integers such that $$(x+1)^{y+1}+1=(x+2)^{z+1}.$$

\item (1999 Bulgaria) Prove that the equation
$$x^3+y^3+z^3+t^3=1999$$
has infinitely many integer solutions.

\item (1995 Romanian training) Find all integer and positive
solutions $(x,y,z,t)$ of the equation
$$(x+y)(y+z)(z+x)=txyz$$
such that $(x,y)=(y,z)=(z,x)=1.$

\item Find the smallest natural number $n$ such that the sum of
the squares of its divisors (including 1 and $n$) equals
$(n+3)^2.$

\item The positive integers $a$, $b$, $x$ satisfy $x^{a+b}=a^bb$.
Show that $a=x$ and $b=x^x$.

\item (1986 IMO Q1) Let $d$ be any positive integer not equal to
2,5 or 13. Show that one can find distinct $a,b$ in the set
$\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

\item (1994 IMO Q4) Determine all ordered pairs $(m,n)$ of
positive integers such that
$$\frac{n^3+1}{mn-1}$$ is an integer.

\item (1997 IMO Q5) Find all pairs $(a,b)$ of integers $a,b\geq 1$
that satisfy the equation $$a^{b^2}=b^a.$$

\item (1998 IMO Q4) %% IMO4, UK
Determine all pairs $(a, b)$ of positive integers such that
$ab^2+b+7$ divides $a^2 b+a+b$.


\item (2001 IMO Q6) Let $a$, $b$, $c$, $d$ be integers with
$a>b>c>d>0$. Suppose that $$ac+bd=(b+d+a-c)(b+d-a+c).$$ Prove that
$ab+cd$ is not prime.

\item Let $p$ be a prime. Suppose that there exist integers
$a,b,x,y$ such that $p=a^2+b^2=x^2+y^2$. Show that
$\{a,b\}=\{x,y\}$.

\item Show that the following equation has infinitely many
solutions in natural numbers with $(x,y,z,t)=1$:
$$x^3+y^3=z^4-t^2$$

\end{enumerate}


\end{document}
