\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 21-st Austrian--Polish Mathematics Competition}\\
Poland, June 29 -- July 1, 1998
\end{center}

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

\item
Let $x_1,x_2,y_1,y_2$ be real numbers such that $x_1^2+x_2^2\leq 1\,.$
Prove the inequality
$$(x_1y_1+x_2y_2-1)^2 \geq (x_1^2+x_2^2-1)(y_1^2+y_2^2-1)\,.$$

\item
Consider $n$ points $P_1,P_2,\ldots ,P_n$ lying in that order on a straight
line. We colour each of these $n$ points using one of the following
colours: white, red, green, blue and violet. A colouring is called admissible
if for each two consecutive points $P_i, P_{i+1}$ $(i=1,2,\ldots ,n-1)$
either both are of the same colour or at least one of them is white. How many
admissible colourings are there?

\item
Find all pairs of real numbers $(x,y)$ satisfying the following system of
equations
$$ 2-x^3=y\,,\;\;2-y^3=x\,.$$

\item
Let $m,n$ be positive integers. Denote
$$S_m(n)=\sum_{1\leq k\leq n}\Big[\sqrt[k^2]{k^m}\Big]$$
($[x]$ is the biggest integer not bigger than $x$). Prove, that
$$S_m(n)\leq n+m\cdot \big(\sqrt[4]{2^m}-1\big)\,.$$

\item
Determine all pairs $(a,b)$ of positive integers such that the equation
$$x^3-17x^2+ax-b^2=0$$
has three integer roots (not necessarily different).

\item
Different points $A, B, C, D, E, F$ lie on the circle $k$ in that order.
The tangents to the circle $k$ in the points $A$ and $D$ and the lines
$BF$ and $CE$ intersect in one point $P$. Prove that the lines 
$AD$, $BC$ and $EF$ are either parallel or intersect in one point.

%\clearpage
\item

Consider all pairs $(a,b)$ of natural numbers such that the product
$a^a\cdot b^b$ written in decimal system ends with exactly $98$ zeros.
Find the pair $(a,b)$ for which the product $ab$ is the smallest.

\item
Let $n>2$ be a given natural number. Consider a square net on the plain.
In each unit square of the net a natural number is written. The polygons
with area equal to $n$, whose sides are contained in the lines forming the net,
are called admissible. The sum of all the numbers written in the
squares contained in a polygon is called value of the polygon. Prove that if 
the values of any two congruent admissible polygons are equal
then all the numbers written in the unit squares of the net are equal.\\
Remark. We recall that the symmetric image $Q$ of the polygon $P$
is congruent to $P$.

\item
Let $K,L,M$ be the midpoints of the sides $BC,AC,AB$ of triangle $ABC$.
The points  $A,B,C$ divide the circumcircle of triangle $ABC$
on three arcs $\widehat{AB}, \widehat{BC}, \widehat{CA}$. Let $X$ be the point
of the arc $\widehat{BC}$ such that $BX=XC$. Analogously, let $Y$ be the point
of the arc $\widehat{AC}$ such that $AY=YC$ and $Z$ be the point of the arc
$\widehat{AB}$ such that $AZ=ZB$. Denote by $R$ the circumradius of triangle $ABC$
and by $r$ the inradius of triangle $ABC$. Prove that
$$ r+KX+LY+MZ=2R\,.$$
\end{list}

\end{document}

