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\begin{center}
{\large\bf 50th Mathematical Olympiad in Poland}\\[.4ex]
{\large Problems of the first round, September -- December 1998}
\end{center}

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\item  %1
Prove that among the numbers of the form $50^n+(50n+1)^{50}$,
where $n$ is a natural number, there exist infinitely many 
composite numbers.

\item  %2
Show that for all real numbers $a$, $b$, $c$, $d$, 
the following inequality holds:
$$(a+b+c+d)^2\leq 3(a^2+b^2+c^2+d^2)+6ab\,.$$

\item  %3
In the isosceles triangle $ABC$ the angle $BAC$ is a right angle.
Point $D$ lies on the side $BC$ and satisfies $BD=2\cdot CD$.
Point $E$ is the foot of the perpendicular of the point $B$
on the line $AD$. Find the angle $CED$.

\item  %4
Let $x$, $y$ be real numbers such that the numbers
$x+y$, $x^2+y^2$, $x^3+y^3$ and $x^4+y^4$ 
are integers. Prove that for all positive integers
$n$, the number $x^n+y^n$ is an integer.

\item  %5
Find all pairs of positive integers $x$, $y$ satisfying the equation
%$x^{\mbox{\small $y$}}=y^{50}$.
$\ {y\,}^{\mbox{\small $\!x$}}=x^{50}\,$.

\item  %6
Diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$
meet at the point $P$. The point $M$ is the midpoint 
of the side $AB$. The line $MP$ meets the side $CD$
at the point $Q$. Prove that the ratio of the areas
of the triangles $BCP$ and $ADP$ equals the ratio
of the lengths of $CQ$ and $DQ$.

\item  %7
Let $n\geq2$ be a positive integer. Find all
polynomials $P(x)=a_0+a_1x+\ldots+a_nx^n$ having
exactly $n$ roots not greater than $-1$
and satisfying 
$$a_0^2+a_1a_n=a_n^2+a_0a_{n-1}\,.$$

\item  %8
Let $n\geq2$ be a positive integer and 
$S$ be a set containing $n$-elements.
Find the smallest positive integer $k$
for which there exist subsets $A_1,A_2,\ldots,A_k$
of the set $S$ with the following property:\\
for any two different elements $a,b\in S$,
there exists a number $j\in\{1,2,\ldots,k\}$, such that
$A_j\cap\{a,b\}$ has exactly one element.

\item  %9
Points $D$, $E$, $F$ lie on the sides $BC$,
$CA$, $AB$ of the triangle $ABC$ respectively. 
The incircles of the triangles $AEF$, $BFD$, $CDE$
are tangent to the incircle of the triangle $DEF$.
Prove that the lines $AD$, $BE$, $CF$ are concurrent.

\item  %10
Let $x_1>0$ be a given real number. The sequence $(x_n)$
is defined by the formula:
$$x_{n+1}=x_n+\frac{1}{x_n^2}\,,\qquad\mbox{for }\ n=1,2,3,\ldots\ .$$
Prove that the limit
$\;\displaystyle{\lim\limits_{n\to\infty}
\frac{x_n}{\sqrt[\mbox{\scriptsize $3$}]{n}}}\;$
exists and find it.

\item  %11
There are two balls in an urn: one white and one black.
We have Moreover we have $50$ white and $50$ black balls.
We repeat $50$ times the following procedure:
we take out one ball from the urn at random and next
we put it into the urn again together with another ball
of the same colour. Finally we have $52$ balls in the urn.
What is the most probable number of white balls in the urn?

\item  %12
All vertices of the cube with edge length $a$ lie on 
the surface of the regular tetrahedron with edge length $1$.
Find all possible values of $a$.

\end{list}
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