\documentclass[11pt]{article}
\usepackage{geometry} % See geometry.pdf to learn the layout options. There are lots.
\geometry{a4paper} % ... or a4paper or a5paper or ...
%\geometry{landscape} % Activate for for rotated page geometry
%\usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}
\usepackage{enumerate}
\title{Report on the 3rd European Girls' Mathematical Olympiad \\ 10th
-- 16th April 2014 \\ Antalya, Turkey}
\author{Hannah Roberts}
\date{} % Activate to display a given date or no date
\begin{document}
\maketitle{}
\section*{Introduction}
The European Girls' Mathematical Olympiad is now in its third
year. The competition aims to
give a large number of female students the experience of competing in
an international event and hence encourage the participation of girls
in olympiad mathematics. Having been based in previous years in Murray Edwards College, Cambridge, and a Youth Hostel
in Luxembourg City this third edition was located in a 5* all-inclusive hotel on
the Turkish Riviera! The larger budget than in previous years allowed
the inclusion of 29 teams (an increase on last year's 22), with 7 of
those being entries from guest countries. The selection for the UK
EGMO team was made from performances at BMO2, the second round of the
British Mathematical Olympiad.
The team consisted of:
\vspace{0.3cm}
\begin{center}
\begin{tabular}{l l l}
UNK1 & Olivia Aaronson & St Paul's Girls' School \\
UNK2 & Katya Richards & School of St Helen and St Katharine \\
UNK3 & Eloise Thuey & Caistor Grammar School \\
UNK4 & Kasia Warburton & Reigate Grammar School
\end{tabular}
\end{center}
\vspace{0.3cm}
The reserve was Alyssa Dayan of Westminster School. The Team Leader
was Hannah Roberts of Pembroke College, Oxford and the Deputy Leader
was Jo Harbour of Mayfield Primary School, Cambridge.
\section*{The Competition}
The format of the EGMO is much like that of the International Mathematical Olympiad (IMO). Two $4\frac{1}{2}$ hour papers each containing three questions (ordered by approximate difficulty) are sat over two days. Each question carries 7 marks, with part marks given only for significant progress, such that it is usually rare to be awarded 3 or 4. The scores for the UK team were as follows:
\vspace{0.3cm}
\begin{center}
\begin{tabular}{l | r r r r r r | r l }
&Q1& Q2& Q3& Q4& Q5& Q6& Total& Medal \\
Olivia Aaronson &0 & 0& 0& 2& 2& 0& 4& \\
Katya Richards &6 &0 & 0& 7& 7& 0& 20& Silver Medal \\
Eloise Thuey &2 & 2& 0& 1& 0& 1& 6& \\
Kasia Warburton & 6& 0& 2& 7& 3& 0& 18&
Silver Medal
\end{tabular}
\end{center}
\vspace{0.3cm}
The medal boundaries were 24 for gold, 16 for silver and 7 for
bronze. The UK came 8th out of 29 participating countries in the
unofficial team competition, our best result yet. Paper 1 was
particularly hard this year and many students scored no marks on the
first day. Hence the medal boundaries reflected this. Even so,
the papers contained a great set of questions, included below.
\section*{The Problems}
\subsection*{Day 1}
\paragraph*{Problem 1.}
Determine all real constants $t$ such that whenever $a$, $b$, $c$ are the lengths of the sides of a triangle, then so are $a^2+bct$, $b^2+cat$, $c^2+abt$.
\paragraph*{Problem 2.}
Let $D$ and $E$ be points in the interiors of sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$. Let the lines $CD$ and $BE$ meet at $F$. Prove that the incentre $I$
of triangle $ABC$, the orthocentre $H$ of triangle $DEF$ and the midpoint $M$ of the arc $BAC$ of the circumcircle of triangle $ABC$ are collinear.
\paragraph*{Problem 3.}
We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n)=k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a$, $b$ satisfying $a+b=n$.
\subsection*{Day 2}
\paragraph*{Problem 4.}
Determine all integers $n\geq 2$ for which there exist integers $x_1,x_2,\dots,x_{n-1}$ satisfying the condition that if $0