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\begin{center}
{\Large\bf 52nd Mathematical Olympiad in Poland}\\[.4ex]
{\Large Problems of the first round, September -- December 2000}
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\item  %1
Solve in integers the equation
$$x^{2000}+2000^{1999}=x^{1999}+2000^{2000}\;.$$

\item  %2
The points $D$ and $E$ lie on the sides $BC$ and $AC$ of the triangle $ABC$,
respectively. The lines $AD$ and $BC$ meet in the point $P$. The points $K$ and $L$
lie on the sides $BC$ and $AC$, respectively and are chosen so that $CLPK$
is a parallelogram. Prove that
 $$\frac{AE}{EL}=\frac{BD}{DK}\,.$$

\item  %3
Find all positive integers $n\ge 2$, such that the inequality
$$x_1x_2+x_2x_3+\ldots+x_{n-1}x_n\le {n-1\over n}
\left(x_1^2+x_2^2+\ldots+x_n^2\right)$$
is satisfied for all positive real numbers $x_1,x_2,\ldots,x_n$.

\item  %4
Prove or disprove: into a cube box with edge $4$, one can put $65$ balls of diameter $1$.

\item  %5
Prove that for all integers $n\ge 2$ and for all prime numbers $p$ the number
$$\,n^{p^p}+p^p\,$$
is composite. 

\item  %6
The integers $a$, $b$, $x$, $y$ satisfy the equality
$$a+b\sqrt{2001}=\left(x+y\sqrt{2001}\right)^{2000}\;.$$
Prove that $a\ge 44b$.

\item  %7
$ABC$ is an isosceles triangle with the angle $\angle A= 90^\circ$.
The points $D$ and $E$ lie on the side $BC$ and $\angle DAE= 45^\circ$.
The circumcircle of the triangle $ADE$ meets the sides $AB$ and $AC$ in the points
$P$ and $Q$, respectively. Prove that  ${BP+CQ=PQ}$.

\item  %8
%\vspace{-2ex}
\begin{minipage}[t]{13cm}
For which positive integers $m$, $n$, can the rectangle of dimensions $m\times n$ be
cut into the pieces congruent to the one at the figure. Each of the little squares at the
figure has side $1$.
\end{minipage}\hspace{2ex}
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\item  %9
Prove that among any  $12$ consecutive integers there exists an integer which is not
equal to the fourth power of an integer.

\item  %10
Prove that each triangle $ABC$ contains an interior point $P$ possessing the following
property:\\
each line passing through $P$ divides the perimeter and the area of the triangle 
$ABC$ in the same ratio.

\item  %11
The sequence  $(c_1,c_2,\ldots,c_n)$ of positive integers is called admissible if
each integer $k\in \{1,2,\ldots,2( c_1+c_2+\dots+c_n)\}$ can be represented in the form
$$k=\sum_{i=1}^na_ic_i\quad{\hbox{with}}\quad a_i\in\{-2,-1,0,1,2\}\,.$$
For each $n$ find
$$\max\left\{\sum_{i=1}^n c_i\,:\;(c_1,c_2,\ldots,c_n)\;\mbox{is admissible}\right\}\,.$$ 
\item  %12
Consider the sequences $x_0,x_1,\ldots,x_{2000}$ of integers satisfying
$$x_0=0\ \ \ {\hbox{and}}\ \ \ |x_n|=|x_{n-1}+1|\qquad{\hbox{for}}\ \
n=1,2,\ldots,2000\;.$$ 
Find the minimum value of the expression
$\;\left|x_1+x_2+\ldots+x_{2000}\right|\,.$
\end{list}
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