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\begin{center}
{\bf 48-th Mathematical Olympiad in Poland}\\
First Round, September--December, 1996
\end{center}

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

\item  
Solve the system of equations:
$$\left\{
\begin{array}{rcl}
x\cdot |x|+y\cdot |y|&=&1\\
\ [x]+[y]&=&1\,.
\end{array}
\right.$$
{\it Note:\/} $[t]$ denotes the biggest integer not bigger than $t$.

\item
Point $P$ is an interior point of a parallelogram $ABCD$, such 
$\angle ABP=\angle ADP$. Prove that $\angle PAB=\angle PCB$.

\item
Prove that for any real numbers $a,b\geq 1$, $c\geq 0$ and 
for any integer $n\geq 1$ the inequality
$(ab+c)^n-c\leq ((b+c)^n-c)a^n$ holds.

\item 
Prove that a natural number $n\geq 2$ is composite
if and only if there exist natural numbers $a,b,x,y\geq 1$
with $a+b=n$ and $\,\displaystyle{\frac{x}{a}+\frac{y}{b}=1}\,$.

\item 
The angle bisectors of the internal angles $A$, $B$, $C$
of triangle $ABC$ intersect the opposite sides at points
$D$, $E$, $F$, respectively and the circumcircle of $ABC$
at $K$, $L$, $M$, respectively. Prove that
$$\frac{AD}{DK}+\frac{BE}{EL}+\frac{CF}{FM}\geq 9\,.$$

\item
Polynomial $P(x)$ of degree $n$ satisfies the following condition:
$$P(k)=\frac{1}{k}\qquad\mbox{for}\ \ k=1,2,4,8,\ldots,2^n\,.$$
Find $P(0)$.

\item 
Find the supremum of volumes of tetrahedra contained 
in a ball of given radius, whose one edge
is a diameter of the ball.

\item 
Let $a_n$ denote the number of all nonempty subsets
of $\{1,2,\ldots,6n\}$, whose sum of elements gives the
remainder 5 upon division by 6; and let $b_n$ be the number of all
nonempty subsets of $\{1,2,\ldots,7n\}$, whose product of elements
gives the remainder 5 upon division by 7. Find the ratio $a_n/b_n$.

\item 
Determine all functions $f\colon\, [1;\infty)\to [1;\infty)$
satisfying the following two conditions:

\smallskip
\centerline{(i)
$\displaystyle{f(x+1)=\frac{(f(x))^2-1}{x}}\,$ for $x\geq 1$;
\hspace{1cm}(ii) the function $g(x)=f(x)/x$ is bounded.}

\item
Points $P$, $Q$ lie inside an acute triangle $ABC$ and satisfy
$\angle ACP=\angle BCQ$ and $\angle CAP=\angle BAQ$. Points
$D$, $E$, $F$ are the orthogonal projections from $P$
onto the lines $BC$, $CA$, $AB$, respectively. Prove that
the angle $DEF$ is right if and only if $Q$ is the orthocenter
of triangle $BDF$.

\item
Given a natural number $m\geq 1$ and a polynomial $P(x)$
of positive degree with integer coefficients. Prove that
if $P(x)$ has at least three distinct integer roots, then
$P(x)+5^m$ has at most one integer root.

\item
A group consisting of $n$ people noticed that, for some period
of time, three of them might be going for a dinner together,
each pair meeting at exactly one dinner. Prove that
$n$ gives the remainder 1 or 3 upon division by 6.

\end{list}

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