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{\large\bf 49th Mathematical Olympiad in Poland}\\[.4ex]
Problems of the first round, September -- December 1997
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\item
Solve the system of equations:
$$\cases{\ \ \ |x-y| - {\displaystyle{|x|\over x}} = -1 \cr 
|2x-y| + |x+y-1| + |x-y| +y-1=0.\cr}$$

\item
The lines containing the sides of triangle $ABC$, inscribed
in a circle of center $O$, intersect at $H$. It is known
that ${AO=AH}$. Find the measure of the angle $CAB$.

\item
The sequences $(a_n)$, $(b_n)$, $(c_n)$ are given by the conditions:
$a_1 = 4$, $a_{n+1} = a_n(a_n-1)$, $2^{b_n} = a_n$, $2^{n-c_n}= b_n$
for $n = 1,2,3,\ldots$ . Prove that the sequence $(c_n)$ is
bounded.

\item
Given a positive number $a$. Determine all real numbers
$c$ with the following property:
for any pair of positive numbers $x$, $y$ the following
inequality holds
$$(c-1)x^{a+1} \leq (cy-x)y^a.$$ 

\item
Given is an integer $n\geq 1$. Solve the equation:
$$|\tan^nx-\cot^nx| = 2n|\cot 2x|.$$

\item
In triangle $ABC$ with $AB&gt;AC$, 
$D$ is the midpoint of the side $BC$;
$E$ lies on the side $AC$. Points $P$ and $Q$ are the feet
of the perpendiculars from $B$ and $E$, respectively to
the line $AD$. Prove that $BE=AE+AC$ if and only if $AD=PQ$.
 
\item
Given positive integers $m$, $n$. Set $A = \{1,2,3,\ldots,n\}$.
Determine the number of functions $f\colon A \to A$ 
attaining exactly $m$ values and satisfying the condition
$$\mbox{if }\  k,\ell\in A,\  k\leq\ell,\ \mbox{ then }
\ f(f(k))=f(k)\leq f(\ell).$$

\item
Determine if there exists a convex polyhedron having
exactly $k$ edges and a plane not passing through any
of its vertices and cutting $r$ edges with $3r &gt; 2k$.

\item
Let $\ a_0 = 0{,}91\ $ and $\ a_k = 0{,}\underbrace{99\ldots 9}_{2^k}
\underbrace{00\ldots 0}_{2^k-1}1\ $ for $\ k =
1,2,3,\ldots$ . Compute
$$\ \displaystyle{\lim_{n\to \infty} (a_0a_1 \ldots a_n).}$$

\item
The medians $AD$, $BE$, $CF$ of triangle $ABC$ intersect
at $G$. The quadrilaterals $AFGE$ and $BDGF$ are cyclic. 
Prove that the triangle $ABC$ is equilateral. 

\item
In a tennis tournament $n$ players take part.
Everyone has played against everyone else, there were
no draws. Prove that there exists a player $A$
who won with every player $B$ directly or indirectly, i.e.
either $A$ won with $B$ or $A$ won with some
player $C$, who had won with $B$.

\item
Let $g(k)$ be the biggest prime divisor of an
integer $k$ if $|k|\geq 2$, and let $g(-1)=g(0)=g(1)=1.$ 
Determine if there exists a polynomial $W$ of positive degree
with integer coefficients for which the set of the numbers
of the form $g\bigl(W(x)\bigr)$ ($x$ --- an integer) 
is finite.

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