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\begin{center}
{\bf 45-th Mathematical Olympiad in Poland}\\
First Round, September--December, 1993
\end{center}

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

\item Prove that the system of equations
$$\left\{
\begin{array}{l}
a^2-b=c^2\\
b^2-a=d^2
\end{array}
\right.
$$
has no integer solutions $a$, $b$, $c$, $d$.

\item The sequence of functions $f_0,f_1,f_2,\ldots$ is given by the 
conditions:
\begin{eqnarray*}
f_0(x)&amp;=&amp;|x| \qquad \mbox{\rm for all}\ \ x\in {\bf R}\\
f_{n+1}(x)&amp;=&amp;|f_{n}(x)-2|\qquad \mbox{\rm for}\ \ n=0,1,2,\ldots\quad
\mbox{\rm and all}\ \ x\in {\bf R}\,.\\[-3ex]
\end{eqnarray*}
For each positive integer $n$, solve the equation $f_n(x)=1$.

\item 
Prove that if $a$, $b$, $c$ are the lengths of the sides of a triangle,
then
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\leq
\frac{1}{a+b-c}+\frac{1}{c+a-b}+\frac{1}{b+c-a}\,.$$

\item 
Given is a circle with center $O$, point $A$ inside the circle
and a chord $PQ$ which is not a diameter and passing through
$A$. The lines $p$ and $q$ are tangent to the given circle at $P$ and
$Q$ respectively. The line $\ell$ passing through $A$ and perpendicular
to $OA$ intersects the lines $p$ and $q$ at $K$ and $L$ respectively.
Prove that $|AK|=|AL|$.

\item 
Prove that if the polynomial $x^3+ax^2+bx+c$ has three distinct
real roots, so does the polynomial
$$x^3+ax^2+\frac{1}{4}(a^2+b)x+\frac{1}{8}(ab-c)\,.$$

\item 
The function $f\colon\,{\bf R}\to {\bf R}$ is continuous.
Prove that if for every real number $x$, there exists a positive
integer $n$, such that
$$(\underbrace{f\circ f\circ \ldots\circ f}_{n})(x)=1\,,$$
then $f(1)=1$.

\item 
Given convex quadrilateral $ABCD$. We construct the similar triangles
$APB$, $BQC$, $CRD$, $DSA$ outside $ABCD$ so that
$$\angle PAB=\angle QBC=\angle RCD=\angle SDA\,,\qquad
\angle PBA=\angle QCB=\angle RDC=\angle SAD\,.$$
Prove that if $PQRS$ is a parallelogram, so is $ABCD$.

\item 
Given positive integers $a$, $b$, $c$ such that $a^3$ is divisible
by $b$, $b^3$ is divisible by $c$, $c^3$ is divisible by $a$.
Prove that $(a+b+c)^{13}$ is divisible by $abc$.

\item 
In a conference $2n$ personalities take part. Each person 
has at least $n$  acquaintaces among the others. Prove that
it is possible to quarter the participants 
into two-person rooms, so that each participant would share the
room with his/her acquaintace.

\item 
Given positive real numbers $p$, $q$ with $p+q=1$. Prove that for
all positive integers $m$, $n$ the following inequality holds
$$(1-p^m)^n+(1-q^n)^m\geq 1\,.$$

\item 
A triangle with perimeter $2p$ is inscribed in a circle of radius
$R$ and also circumscribed on a circle of radius $r$. Prove that
$p&lt;2(R+r)$.

\item Prove that the sums of the opposite dihedral angles 
of a tetrahedron are equal if and only if the sums of 
the the opposite edges of this tetrahedron are equal.

\end{list}

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