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\begin{center}
{\bf 49-th Mathematical Olympiad in Poland}\\
Second Round, February 27--28, 1998
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\begin{list}{\bf\arabic{zadanie}.}{\usecounter{zadanie}}

\centerline{\bf First Day}

\item  
Let $A_n={1,2,\ldots ,n}$. Prove or disprove the following statement:\\
for all integers $n\geq 2$ there exist functions $f:A_n\rightarrow A_n$
and $g:A_n\rightarrow A_n$ which satisfy\\
$f(f(k))=g(g(k))=k$ for $k=1,2,\ldots ,n\,$;\\
$g(f(k))=k+1$ for $k=1,2,\ldots ,n-1\,.$

\item 
In triangle $ABC$ the angle $\angle BCA$ is obtuse
and $\angle BAC = 2\angle ABC\,.$ The line through $B$ and perpendicular to 
$BC$ intersects line $AC$ in $D$. Let $M$ be the midpoint of $AB$. Prove that
$\angle AMC=\angle BMD$.

\item
a) Assume that nonnegative numbers $a\,,b\,,c\,,d\,,e\,,f$ with sum equal to $1$ satisfy
$$ ace+bdf \geq \frac{1}{108}\,.$$ 
Show that
$$abc+bcd+cde+def+efa+fab\leq \frac{1}{36}\,.$$
b) Do there exist six different positive numbers $a\,,b\,,c\,,d\,,e\,,f$ 
with sum equal to $1$ for which the two above inequalities become equalities? 

\vspace{20pt}

\centerline{\bf Second Day} 

\item
Find all pairs $(x,y)$ of integers which satisfy the equation
$x^2+3y^2=1998x$.

\item
Suppose that nonnegative numbers $a_1,a_2,\ldots ,a_7,b_1,b_2,\ldots ,b_7$ satisfy
$$a_i+b_i\leq 2 \quad \mbox{{\rm for}}\;\,i=1,2,\ldots ,7\,.$$
Prove that there exist two different indices $k,m\in \{1,2,\ldots ,7\}$
for which 
$$|a_k-a_m|+|b_k-b_m|\leq 1\,.$$

\item
Prove that in tetrahedron $ABCD$ the edge $AB$ is perpendicular to the edge $CD$ if
and only if there exists a parallelogram $CDPQ$ such that $PA=PB=PD$ and $QA=QB=QC$.

\end{list}

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