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\title{Problem Sheet 2}
\author{UK IMO squad training camp, Bath 2001}
\date{}
\begin{document}
\maketitle
\begin{enumerate}
\item (Israel) Find all solutions of the pair of simultaneous equations
\[\left\{
\begin{array}{rcl}
x_1 + x_2 + \cdots + x_{2000} & = & 2000\\
x_1^4 + x_2^4 + \cdots + x_{2000}^4& = & x_1^3 + x_2^3 + 
\cdots + x_{2000}^3
\end{array}\right.
\]
\item (Israel) Given 2001 real numbers $x_1, x_2, \ldots, x_{2001}$
such that $0 \leq x_n \leq 1$ for each $n = 1,2, \ldots, 2001,$ find the 
maximum value of 
\[
\left( \frac{1}{2001} \sum_{n=1}^{2001} x_n^2 \right)
- \left( \frac{1}{2001} \sum_{n=1}^{2001} x_n \right)^2.\]
Where is this maximum attained?
\item (Hungary) Let $t$ and $r$ be the area and the inradius of the triangle
$ABC$. Let $r_a$ denote the radius of the circle touching the incircle, $AB$
and $AC$. Define $r_b$ and $r_c$ in a similar fashion. The common tangent of the circles with radii $r$ and $r_a$ cuts a little triangle from $ABC$ with area 
$t_a$. Quantities $t_b$ and $t_c$ are defined in a similar fashion. 
Prove that
\[ \frac{t_a}{r_a} +
 \frac{t_b}{r_b} +
 \frac{t_c}{r_c} = \frac{t}{r}.\]
\item (Hungary-Israel competition) In this question, $G_n$ 
is a simple undirected graph with $n$ vertices, $K_n$ is the 
complete graph with $n$ vertices, and $e(G_n)$ is the number of edges
of the graph $G_n$. Let $C_n$ denote circle with $n$ verticies.
\begin{enumerate}
\item[(i)] The edges of $K_n$, $n \geq 3$ are coloured with $n$ colours, and 
every colour appears at least once. Prove that one can find a 
triangle whose sides are coloured with 3 different colours.
\item[(ii)] Suppose that $n \geq 5$ is given. If $e(G_n) \geq n^2/4 + 2$, prove that there exist
two triangles which have exactly one common vertex.
\item[(iii)] Suppose that $e(G_n) \geq \frac{n \sqrt n}{\sqrt 2} + n$.
Prove that $G_n$ contains $C_4$.
\end{enumerate}
\item (Israel) 2001 lines are given in the plane. No two of them are 
parallel and no three meet in a point.  These lines partition the plane into 
some {\em regions} (not necessarily finite) bounded by segments of these lines. 
is called a {\em map}. Two regions on the map are called {\em neighbours} if 
they share a side. The set of 
intersection points of lines is called the set of {\em verticies}. Two vertices
are called neighbours if they are found on the same side.

A legal colouring of the regions (one colour per region) must have
neighbouring regions coloured differently.
A legal colouring of the vertices (one colour per vertex) must have
neighbouring vertices coloured differently. 
\begin{enumerate}
\item[(i)] What is the minimum number of colours required for the legal 
colouring of any map?
\item[(ii)] What is the minimum number of colours required for the legal 
colouring of the vertices of any map?
\end{enumerate}
\item (Israel) Triangle $ABC$ in the plane $\Pi$ is said to be {\em good}
if it has the following property: for any point $D$ in space, out of the plane 
$\Pi$, it is possible to construct a triangle with sides of lengths 
$|AD|, |BD|$ and $|CD|$. Find all the good triangles.
\item (Israel) Find a pair of integers $(x,y)$ such that 
$15x^2 + y^2 = 2^{2000}$. Does there exist such a pair $(x,y)$ with
$x$ odd?
\end{enumerate}
\end{document}