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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 25--26, 2000}
\end{center}

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

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

\item
Prove or disprove the following statement:\\
Each positive rational number can be written in the form
$$ \frac{a^2+b^3}{c^5+d^7}\,$$
where $a,\,b,\,c,\,d$ are positive integers.

\item
In the triangle $ABC$ the bisector of the angle $\angle BAC$
meets the circumcircle of the triangle $ABC$
in the point $D\neq A$. The points $K$ and $L$
are the perpendicular projections of the points $B$ and $C$
onto the line $AC$, respectively. Prove that
$$ AD \geq BK + CL\,.$$

\item
In the unit squares of the $n\times n$ chessboard are written
$n^2$ different positive integers. In each column of the chessboard
the unit square with the greatest number is coloured red. A set
$S$ of $n$ unit squares is called admissible if any two unit squares
from $S$ do not lie in the same column or in the same row of the
chessboard.
Prove that the admissible set with the greatest sum of numbers written
in its unit squares contains at least one red unit square.

\vspace{20pt}

\centerline{\bf Second Day}


\item
Point $I$ is the incentre of the triangle $ABC$ with $AB\neq AC$.
The lines $BI$ and $CI$ meet the sides
$AC$ and $AB$ in the points $D$ and $E$ respectively. Find all
angles $\angle BAC$ for which the equality $DI=EI$
can be satisfied.

\item
Let $N$ denote the set of all positive integers. Prove or disprove that:

There exists a function $f:N\rightarrow N$ such that
the equality $f(f(n))=2n$ holds for all $n\in N$.

\item
Let $w$ be a polynomial of degree two with integer coefficients.
Suppose that for each integer $x$ the value $w(x)$ is the square
of an integer. Prove that $w$ is the square of a polynomial.
\end{list}

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