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\begin{center}
{\bf 47-th Mathematical Olympiad in Poland}\\
First Round, September--December, 1995
\end{center}

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

\item Determine all positive integers $n$, such that the
equation $2\sin nx=\tan x+\cot x$ has solutions in real
numbers $x$.

\item A number is called a palindromic number if its
decimal representation read from the left to the right
is the same as read from the right to the left. Let $(x_n)$
be the increasing sequence of all palindromic numbers.
Determine all primes, which are the divisors of at least
one of the differences $x_{n+1}-x_n$.

\item In a group of $kn$ persons, each person knows more
than $(k-1)n$ others ($k$, $n$ are positive integers).
Prove that one can choose $k+1$ persons from this
group so that each chosen person knows all the others chosen.

{\it Note:\/} If a person $A$ knows $B$, then $B$ knows $A$.

\item A line tangent to the incircle of the equilateral
triangle $ABC$ intersects the sides $AB$ and $AC$ at points
$D$ and $E$ respectively. Prove that
$$\frac{AD}{DB}+\frac{AE}{EC}=1\,.$$

\item Given triangle $ABC$ in the plane such that $\angle CAB=
\alpha&gt;\pi/2$. Let $PQ$ be a segment whose midpoint
is the point $A$. Prove that
$$(BP+CQ)\tan\frac{\alpha}{2}\geq BC\,.$$

\item Given two sequences of positive integers: the
arithmetic sequence with difference $r&gt;0$ and the geometric
sequence with ratio $q&gt;1$; $r$ and $q$ are coprime.
Prove that if these sequences have one term in common, then
they have them infinitely many.

\item Nonnegative numbers $a$, $b$, $c$, $p$, $q$, $r$
satisfy the conditions:
$$a+b+c=p+q+r=1; \qquad p,q,r\leq\frac{1}{2}\,.$$
Prove that $8abc\leq pa+qb+rc$ and determine when
equality holds.

\item The ray of light starts from the center of a square
and reflects from its sides with the principle that {\it
the angle of reflection is equal to the angle of incidence\/}.
After some time the ray returns to the center of the
square. The ray never reached the vertex and has never returned to
the center of the square before. Prove that the ray reflected
from the sides of the square an odd number of times.

\item A polynomial with integer coefficients when divided
by $x^2-12x+11$ gives the reminder $990x-889$. Prove that
the polynomial has no integer roots.

\item Prove that the equation $x^x=y^3+z^3$ has infinitely
many solutions in positive integers $x$, $y$, $z$.

\item In a skiing jump competition 65 contestants take part.
They jump with the previously established order. Each of
them jumps once. We assume that the obtained results are
different and the orders of the contestants after the
competition are equaly likely. In each moment of the
competition by a leader we call a person who is scored
the best at this moment. Denote by $p$ the probability
that during the whole competition there was exactly one
change of the leader. Prove that $p&gt;1/16$.

\item Find out whether there exist two congruent cubes
with a common center such that each face of one
cube has a common point with each face of the other.

\end{list}

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