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\begin{center}
{\bf 46-th Mathematical Olympiad in Poland}\\
First Round, September--December, 1994
\end{center}

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

\item Determine all pairs $(x,y)$ of natural numbers,
such that the numbers $\displaystyle{\frac{x+1}{y}}$ and
$\displaystyle{\frac{y+1}{x}}$ are natural. 

\item Given a positive integer $n\geq 2$. Solve the following system of 
equations: $$\left\{
\begin{array}{l}
x_1|x_1|=x_2|x_2|+(x_1-1)|x_1-1|\\
x_2|x_2|=x_3|x_3|+(x_2-1)|x_2-1|\\
\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\
x_n|x_n|=x_1|x_1|+(x_n-1)|x_n-1|\,.
\end{array}
\right.
$$

\item A quadrilateral with sides $a$, $b$, $c$, $d$ is inscribed
in a circle of radius $R$. Prove that if $a^2+b^2+c^2+d^2=8R^2$,
then either one of the angles of the quadrilateral is right 
or the diagonals of the quadrilateral are perpendicular.

\item In some school $64$ students participate in five different
subject olympiads. In each olympiad at least $19$ students take
part; none of them is a participant of more than three olympiads.
Prove that if every three olympiads have a common paticipant,
then there are two olympiads having at least five participants
in common.

\item Given positive numbers $a$, $b$. Prove that the following
sentences are equivalent:

(1) $\sqrt{\vphantom{b}a}+1>\sqrt{b}$;

(2) for every $x>1$, $\displaystyle{ax+\frac{x}{x-1}>b}$.

\item Inside triangle $ABC$ there is chosen a point $P$.
The rays $AP$, $BP$, $CP$ intersect the boundary of the triangle
in the points $A'$, $B'$, $C'$ respectively. Set
$$u=|AP|:|PA'|,\quad v=|BP|:|PB'|,\quad w=|CP|:|PC'|\,.$$
Express the product $uvw$ in terms of the sum $u+w+v$. 

\item (a) Find out, whether there exists a differentiable function
$f\colon \R\to\R$, not equaling $0$ for all $x\in \R$, satisfying the
conditions $2f(f(x))=f(x)\geq 0$ for all $x\in \R$.

(b) Find out, whether there exists a differentiable function
$f\colon \R\to\R$, not equaling $0$ for all $x\in \R$, satisfying the
conditions $-1\leq 2f(f(x))=f(x)\leq 1$ for all $x\in \R$.

\item In a regular pyramid with a regular $n$-gon
as a base, the dihedral angle between a lateral face
and the base is equal to $\alpha$, and 
the angle between a lateral edge and the base
is equal to $\beta$. Prove that
$$\sin^2\alpha -\sin^2\beta\leq \tg^2\frac{\pi}{2n}\,.$$

\item Let $a$ and $b$ be positive real numbers with the
sum equal to $1$. Prove that if $a^3$ and $b^3$ are rational,
so are $a$ and $b$.

\item Given a line $k$ and three distinct points on it. Each of these
points is the beginning of a pair of the rays -- all the rays lie
on the same side of the halfplane of the edge $k$. Each of these three
pairs form a convex quadrilateral with another pair (so we have
three quadrilaretals formed by these pairs of the rays). Prove that
if it is possible to inscribe a circle in two of these quadrilaterals,
then it is possible to inscribe a circle in the third one as well.

\item Given are natural numbers $n>m>1$. We draw $m$ numbers
from the set $\{1,2,\ldots,n\}$ one by one without putting
the drawn numbers back.
Find the expected value of the difference between the largest
and the smallest of the drawn numbers.

\item The sequence $(x_n)$ is given by
$$x_1=\frac{1}{2},\quad x_n=\frac{2n-3}{2n}\cdot x_{n-1}\quad
\mbox{\rm for}\hspace{0.2cm} n=2,3,\ldots\,.$$
Prove that for all natural numbers $n\geq 1$ the following
inequality holds
$$x_1+x_2+\ldots+x_n<1.$$

\end{list}

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