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\begin{center}
{\large\bf 51th Mathematical Olympiad in Poland}\\[.4ex]
{\large Problems of the first round, September -- December 1999}
\end{center}

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\item  %1
Let $n\geq3$ be a positive integer. Prove that the sum of the cubes 
of all natural numbers, coprime and less than $n$, is divisible by $n$.  

\item  %2
In the acute-angled triangle $ABC$ holds 
$\angle ACB = 2\angle ABC$. The point $D$
lies on the side $BC$ and satisfies $2\angle BAD=\angle ABC$. Prove that
$$\frac{1}{BD}=\frac{1}{AB}+\frac{1}{AC}\,.$$ 

\item  %3
The sum of positive numbers $a$, $b$, $c$ is equal to $1$.
Prove that $a^2+b^2+c^2+2\sqrt{3abc} \leq 1.$

\item  %4
Each point of a circle is painted with one of three colours.  
Prove that there exist three points on the circle which have 
the same colour and are vertices of an isosceles triangle. 

\item  %5
Find all pairs $(a,b)$ of positive integers for which
the numbers $a^3+6ab+1$ and $b^3+6ab+1$ are cubes of positive integers.

\item  %6
A point $X$ lies inside or on the boundary of the triangle $ABC$ in which
the angle $\angle C$ is a right angle. The points $P, Q$ and $R$
are the perpendicular projections of $X$ onto the sides $BC$, $CA$ 
and~$AB$ respectively. Prove that the equality 
${AR\cdot RB=BP\cdot PC+AQ\cdot QC}$ holds
if and only if the point $X$ lies on the side $AB$.

\item  %7
Prove that for each positive integer $n$ and each number $t \in
({1\over2},1)$ there exist numbers $a, b \in (1999,2000)$,
such that
$${1\over2}a^n + {1\over2} b^n < \bigl( ta + (1-t)b \bigr) ^n.$$ 

\item  %8
The numbers $c(n,k)$ are defined for nonnegative integers
${n\ge k}$ in such a way, that the following equalities hold:
$$\displaylines{c(n,0)=c(n,n)=1\qquad\hbox{for all}\;\;\;n\ge0,\cr
c(n{+}1,k)=2^kc(n,k)+c(n,k{-}1)
\quad\ \hbox{for}\;\;\;n\ge k\ge1.\cr}$$
Prove that $\;{c(n,k)=c(n,n{-}k)}\;$ for ${n\ge k\ge0}$.

\item  %9
Let $m$ and $n$ be given positive integers such that
$m n \vert m^2+n^2+m$. Prove that $m$ is a square of an integer.

\item  %10
Let $\overrightarrow{OA}$, $\overrightarrow{OB}$,
$\overrightarrow{OC}$ be pairwise perpendicular 
unit vectors in three dimensional space. Let $\omega$ be a plane containing
the point $O$, and $A'$, $B'$, $C'$ be the perpendicular projections 
of the points $A$, $B$, $C$ onto the plane $\omega$.
Find the set of numbers $OA'^2 +  OB'^2 + OC'^2$ for
all planes $\omega$.

\item  %11
Let $m$ be a positive integer and $M$ be a set containing
$\,{n^2{+}1}$ positive integers and having the following property:
in each $n+1$ numbers from the set $M$ there are two numbers
for which one is divisible by the other. Prove that the set $M$ 
contains different numbers $a_1,\ldots,a_{n+1}$ for which
$a_{i+1} \vert a_i$ for ${i=1,\ldots,n}$.

\item  %12
In the acute-angled triangle $ABC$ the points
$D$, $E$, $F$ lie on the sides $BC$, $CA$, $AB$
respectively. The circumcircles of the triangles
$AEF$, $BFD$, $CDE$ meet in the point $P$. 
Prove that if
$${{PD}\over{PE}}={{BD}\over{AE}}\,,\qquad 
{{PE}\over{PF}}={{CE}\over{BF}}\,,\qquad   
{{PF}\over{PD}}={{AF}\over{CD}}\, ,$$      
then $AD$, $BE$, $CF$ are the altitudes %wysoko/sciami      
of the triangle $ABC$.

\end{list}
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