\documentstyle[11pt]{article}
\pagestyle{empty}
\textwidth=523pt
\textheight=690pt
\voffset=-90pt %-50pt for A4 paper
\oddsidemargin=-34pt
\evensidemargin=-34pt  

\begin{document}
%\LARGE
\abovedisplayskip=4pt plus 2pt minus 2pt
\belowdisplayskip=4pt plus 2pt minus 3pt

\newcounter{zadanie}
\setcounter{zadanie}{0}

\begin{center}
{\bf The 22nd Austrian--Polish Mathematics Competition}\\
Austria, June 30 -- July 2, 1999
\end{center}

\begin{list}{\bf\arabic{zadanie}.}{\usecounter{zadanie}}

\item
Let $n$ be a positive integer and $M={1,2,\ldots ,n}$. Find the number
of ordered 6-tuples $(A_1,A_2,A_3,A_4,A_5,A_6)$ which satisfy the following
two conditions:\\
a) sets $A_1,A_2,A_3,A_4,A_5,A_6$ (not necessarily different) are subsets of $M$\\
b) each element of $M$ belongs either to exactly three subsets or to exactly six subsets
or does not belong to any subset $A_1,A_2,A_3,A_4,A_5,A_6$.

\item
Find the largest real number $C_1$ and the smallest real number $C_2$ such that
for all real numbers $a,b,c,d,e$ the following inequalities hold
$$C_1<\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a}<C_2\,.$$

\item
Let $n\geq 2$ be a given integer. Determine all systems of $n$ functions $(f_1,\ldots ,f_n)$
where $f_i:R\rightarrow R$ $i=1,\ldots ,n$ such that for all $x,y\in R$ the following
equalities hold
\begin{eqnarray*}
f_1(x)-f_2(x)f_2(y)+f_1(y) & = & 0\\
f_2(x^2)-f_3(x)f_3(y)+f_2(y^2) & = & 0\\
\ldots\ldots & & \\
f_k(x^k)-f_{k+1}(x)f_{k+1}(y)+f_k(y^k) & = & 0\\
\ldots\ldots & & \\
f_n(x^n)-f_1(x)f_1(y)+f_n(y^n) & = & 0\,.
\end{eqnarray*}

\item
Through a point $P$, which lies inside the triangle $ABC$, are
drawn three straight lines $k,l,m$ in such a way that:\\
a) $k$ meets the lines $AB$ and $AC$ in $A_1$ and in $A_2$ ($A_1\neq A_2$)
respectively and $PA_1=PA_2$\,,\\
b) similarly $l$ meets the lines $BC$ and $BA$ in $B_1$ and in $B_2$ ($A_1\neq A_2$)
respectively and $PB_1=PB_2$\,,\\
c) and similarly $m$ meets the lines $CA$ and $CB$ in $C_1$ and in $C_2$ ($C_1\neq C_2$) 
respectively and $PC_1=PC_2$\,.\\
Prove that the lines $k,l,m$ are uniquely determined by the conditions a),b),c).
Find the point $P$ (and prove that there exists exactly one such point)
for which the triangles $AA_1A_2$, $BB_1B_2$, and $CC_1C_2$ have the same area.

\item
A sequence of integers $(a_n)$ satisfies the following recursive equation
$$a_{n+1}=a_n^3+1999\quad\mbox{for}\;\;n=1,2,\ldots\;.$$
Prove that there exists at most one such $n$ for which $a_n$ is the square of an integer.

\item
Solve the following system of equations
\begin{eqnarray*}
x_n^2+x_nx_{n-1}+x_{n-1}^4 = 1 &\mbox{for} &n=1,2,\ldots\,,1999\\
x_0=x_{1999}&&
\end{eqnarray*}
in the set of nonnegative real numbers.
%\clearpage
\item
Find all pairs $(x,y)$ of positive integers such that
$$x^{x+y}=y^{y-x}\,.$$

\item
Let $g$ be a given straight line and let the points $P,Q,R$ all lie on the same side
of the line $g$. The points $M,N$ lie on the line $g$ and satisfy $PM\perp g$ and 
$QN\perp g$. The point $S$ lies between the lines $PM$ and $QN$ and additionaly
satisfies $PM=PS$ and $QN=QS$. The bisectors of $SM$ and $SN$ meet in the point $R$.
The line $RS$ intersects the circumcircle of the triangle $PQR$ in $T\neq R$. Prove
that $S$ is the midpoint of the segment $RT$.

\item
A point in the plane with both integer cartesian coordinates is called a lattice point.
Consider the following one player game. A finite set of selected lattice points and finite set
of selected segments is called a position in this game if the following hold:\\
a) the endpoints of each selected segment are lattice points,\\
b) each selected segment is parallel to a coordinate axis, or to the line $y=x$, or to the
line $y=-x$,\\
c) each selected segment contains exactly five lattice points and all of them
are selected,\\
d) each two selected segments have at most one common point.\\
A move in this game consists of selecting a lattice point and a segment such
that the new set of selected lattice points and selected segment is a position.
Prove or disprove that there exists an initial position such that
the game has infinitely many moves. 
 
\end{list}

\end{document}

