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\begin{document}\large

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\begin{center}
{\large\bf Mathematical Olympiad in Poland}\\[.4ex]
{\large\bf Second Round, February 26--27, 1999}
\end{center}

\newcounter{zadanie}
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\centerline{\bf First Day}

\begin{list}{\bf\arabic{zadanie}.}{\usecounter{zadanie}%
\setlength{\itemsep}{4mm}}

\item
Let $\,f\colon\langle0,1\rangle\to\R\,$ be a given function, such that
$f(\frac{1}{n})=(-1)^n$ for $n=1,2,\ldots$~. Prove that there do not exist
increasing functions $g\colon\langle0,1\rangle\to\R$, $h\colon\langle0,1\rangle\to\R,$
such that $f=g-h$.

\item 
The cube $S$ with edge length $2$ consists of $8$ unit cubes. 
We will call the cube $S$ with one unit cube removed {\it a piece}.
The cube $T$ with edge length $2^n$ consists of $(2^n)^3$ unit cubes.
Prove that if one unit cube is removed from $T$, then 
the remaining solid can be built with pieces.

\item 
The convex quadrilateral $ABCD$ is inscribed in a circle. 
The points $E$ and $F$ lie on the sides $AB$ and $CD$
respectively and satisfy the condition $AE\,{:}\,EB=CF\,{:}\,FD$.
The point $P$ lies on the straight segment $EF$
and satisfies $EP\,{:}\,PF=AB\,{:}\,CD$. Prove that
the ratio of the areas of the triangles $APD$ and $BPC$
does not depend on the choice of the points $E$ and $F$.

\vspace{20pt}

\centerline{\bf Second Day}


\item
Point $P$ lies inside the triangle $ABC$ and satisfies
$\angle PAB=\angle PCA$ and $\angle PAC=\angle PBA$.
The point $O$ is the center of the circumcircle of the triangle $ABC$.
Prove that $\angle APO$ is a right angle  if $O\neq P$.

\item
Let $S=\{1,2,3,4,5\}$. Find out how many functions $f\colon S\to S$
exist with the following property:
$f^{50}(x)=x$ for all $x\in S$. 

{\it Note:\/} $f^{50}(x)=\underbrace{f\circ f\circ\ldots\circ f}_{50}(x)$.

\item
Let the positive integer $k\geq 2$ and the integers $a_1,a_2,\ldots,a_n$
be given and have the following properties:
$$a_1+2^ia_2+3^ia_3+\ldots+n^ia_n=0\,,\qquad
\mbox{for $\ i=1,2,\ldots,k{-}1$.}$$
Prove that the number $\,a_1+2^ka_2+3^ka_3+\ldots+n^ka_n\,$
is divisible by~$k!\,$.

\end{list}

\end{document}
