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\begin{center}
{\bf The 24th Austrian--Polish Mathematics Competition}\\
Austria, June 2001
\end{center}

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

\item
Determine the number of positive integers $a$ for which there exist nonnegative
integers \\$x_0,x_1,\ldots ,x_{2001}$ satisfying
$$a^{x_0}=\sum_{k=1}^{2001}a^{x_k}\,.$$

\item
Let $n$ be a positive integer greater than $2$. Solve in nonnegative real numbers the
following system of equations
$$x_k+x_{k+1}=x^2_{k+2}\,,\quad k=1,2,\ldots ,n\,,$$
where $x_{n+1}=x_1$ and $x_{n+2}=x_2$.

\item
Real numbers $a,b,c$ are lengths of the sides of a triangle. Prove that
$$2&lt;\frac{a+b}{c}+ \frac{b+c}{a}+\frac{c+a}{b}-\frac{a^3+b^3+c^3}{abc}\le 3\,.$$

\item
Prove that if $a,b,c,d$ are lengths of the successive sides of a quadrangle (not necessarily
convex) with the area equal to $S$, then the following inequality holds
$$ S\le \frac{1}{2}(ac+bd)\,.$$
For which quadrangles does the inequality become equality?

\item
The fields of the $8\times 8$ chessboard are numbered from $1$ to $64$ in the following
manner: For $i=1,2,\ldots ,63$ the field numbered by $i+1$ can be 
%attained 
reched from the field numbered by $i$ by one move
of the knight. Let us choose positive real numbers 
$x_1,x_2,\ldots ,x_{64}$. For each white field numbered by $i$ define the number
$y_i=1+x_i^2-\sqrt[3]{x^2_{i-1}x_{i+1}}$ and for each black field numbered by $j$
define the number $y_j=1+x_j^2-\sqrt[3]{x_{j-1}x^2_{j+1}}$ where $x_0=x_{64}$ and
$x_1=x_{65}$. Prove that 
$$\sum_{i=1}^{64}y_i \ge 48\,.$$

\item 
Let $k$ be a fixed positive integer. Consider the sequence defined by
$$ a_0=1\,,\quad a_{n+1}=a_n + [\sqrt[k]{a_n}]\quad n=0,1,\ldots $$
where $[x]$ denotes the greatest integer less than or equal to $x$.
For each $k$ find the set $A_k$ containing all
integer values of the sequence $(\sqrt[k]{a_n})_{n\ge 0}\,.$

\item
Consider the set $A$ containing all positive integers whose decimal expansion
contains no $0$, and whose sum $S(N)$ of the digits divides $N$.\\
a) Prove that there exist infinitely many elements in $A$ whose decimal expansion
contains each digit the same number of times as each other digit.\\
b) Explain that for each positive integer $k$ there exists an element in $A$
having exactly $k$ digits.

\item
The prism with the regular octagonal base and with all edges of the length equal to $1$
is given. The points $M_1,M_2,\ldots ,M_{10}$ are the midpoints of all the faces of the prism.
For the point $P$ from the inside of the prism denote by $P_i$ the intersection point (not equal
to $M_i$) of the line $M_iP$ with the surface of the prism. Assume that the point $P$
is so chosen that all associated with $P$ points $P_i$ do not belong to any edge of the prism
and on each face lies exactly one point $P_i$. Prove that
$$ \sum_{i=1}^{10}\frac{M_iP}{M_iP_i}=5\,.$$  

\item
Let $n&gt;10$ be a positive integer and let $A$ be a set containing $2n$ elements.
The family $\{A_i:i=1,2,\ldots ,m\}$ of subsets of the set $A$ is called
suitable if:
\begin{itemize}
\item
for each $i=1,2,\ldots ,m$ the set $A_i$ contains exactly $n$ elements, 
\item
for all $i\neq j\neq k\neq i$ the set $A_i\cap A_j\cap A_k$ contains at most
one element.
\end{itemize}
For each $n$ determine the length of a maximal suitable family.

\item 
The sequence $a_1,a_2,\ldots ,a_{2010}$ has the following properties:
\begin{itemize}
\item
each sum of the $20$ successive values of the sequence is nonnegative,
\item
$|a_ia_{i+1}|\le 1\quad\;$ for $i=1,2,\ldots ,2009\,.$
\end{itemize}
Determine the maximal value of the expression
$\;\;{\displaystyle \sum_{i=1}^{2010}a_i\,.}$  
\end{list}

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