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\begin{center}
{\bf 44-th Mathematical Olympiad in Poland}\\
First Round, September--December, 1992
\end{center}

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

\item Solve the following equation in real numbers:
$$\frac{(x^2-1)(|x|+1)}{x + \mbox{\rm sgn\,}x} = [x+1].$$

\item Given is a natural number $n \geq 3$. Solve the system of
equations:
$$\left\{
\begin{array}{l}
\tan x_1+3\cot x_1=2\tan x_2 \\
\tan x_2+3\cot x_2=2\tan x_3 \\
\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\
\tan x_n+3\cot x_n=2\tan x_1\,.
\end{array}
\right.
$$

\item Given is a hexagon $ABCDEF$ with a center of symmetry. The lines 
$AB$ and $EF$ meet at the point $A'$, the lines $BC$ and $AF$ meet 
at the point $B'$,
and the lines $AB$ and $CD$ meet at the point $C'$. 
Prove that $AB \cdot BC \cdot CD
=AA' \cdot BB' \cdot CC'$.

\item Determine all functions $f \colon \mbox{\bf R} \longrightarrow
\mbox{\bf R}$ such that 
$$ f(x+y)-f(x-y)=f(x) \cdot f(y) \hspace{0.5cm} \mbox{\rm for} 
\hspace{0.2cm} x,y \in \mbox{\bf R}.$$

\item Given is a halfplane with points $A$ and $C$ on its edge. For every
point $B$ on this halfplane consider the squares $ABKL$ and $BCMN$ lying
outside of the triangle $ABC$. Prove that all the lines $LM$ 
(as the point $B$ varies) have a common point.

\item The sequence $(x_n)$ is determined by the conditions:
$$x_0 = 1992, \hspace{0.5cm} x_n =- \frac{1992}{n} \cdot \sum_{k=0}^{n-1}
x_k \hspace{0.5cm} \mbox{\rm for} \hspace{0.2cm} n \geq 1.$$
Find $\ \displaystyle{\sum_{n=0}^{1992} 2^n x_n\,.}$

\item Given are the points $A_0=(0,0,0)$, $A_1=(1,0,0)$, $A_2=(0,1,0)$,
$A_3=(0,0,1)$ in the space. Let $P_{ij}$ ($i,j \in \{0,1,2,3\}$) be the point
determined by the equality: $\overrightarrow{A_0P_{ij}} = \overrightarrow{%
A_iA_j}$. Find the volume of the smallest convex polyhedron which
contains all the points $P_{ij}$.

\item Given is a positive integer $n \geq 2$. Determine the maximum
value of the sum of natural numbers $k_1,k_2,\ldots,k_n$ satisfying the 
condition: $$k_1^3 + k_2^3 + \cdots + k_n^3\leq 7n\,.$$ 

\item Prove that for all real numbers $a$, $b$, $c$ the inequality
$$ (a^2+b^2-c^2)(b^2+c^2-a^2)(c^2+a^2-b^2)\leq{(a+b-c)}^2{(b+c-a)}^2
{(c+a-b)}^2$$
holds.

\item Let ${\cal C}$ be a cube and let $f\colon {\cal C}\to {\cal C}$
be a surjection with
$$|PQ|\geq |f(P)f(Q)|$$
for all $P,Q\in {\cal C}$. Prove that $f$ is an isometry.

\item Given is an $n\times n$ chessboard. With the same probability, 
we put six pawns on its six cells. Let $p_n$ denotes the
probability that there exists a row or a column containing at 
least two pawns. Find $\displaystyle{\lim_{n\to \infty}np_n}$.

\item Prove that the polynomial $x^n + 4$ can be expressed a product of
two polynomials (each with degree less than $n$) with integer coefficients,
if and only if $n$ is divisible by $4$.

\end{list}

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