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\begin{center}
{\large\bf Problems for the Team Competition Baltic Way 1998}\\
Warsaw, November 8, 1998\\
$4\textstyle{\frac{1}{2}}$ hours.\quad 5 points per problem.
\end{center}

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\def\R{\hbox{\upa \char'122}}     % definition of reals
\def\Q{\hbox{\upa \char'121}}     % rationals
\def\Z{\hbox{\upa \char'132}}     % integers
\def\C{\hbox{\upa \char'103}}     % complex numbers
\def\N{\hbox{\upa \char'116}}     % natural numbers
\def\frac#1#2{{#1\over #2}}

\item%1
Find all functions $f$ of two variables, whose arguments $x$, $y$
and values $f(x,y)$ are positive integers, satisfying the
following conditions (for all positive integers $x$ and~$y$):
\arraycolsep=2pt
$$
\begin{array}{rcl}
f(x,x)&=&x,\\
f(x,y)&=&f(y,x),\\
(x+y)f(x,y)&=&yf(x,x{+}y).
\end{array}
$$


\item%2 
\def\q{quasi-Pythagorean triple}%
\def\Q{Quasi-Pythagorean triple}%
A triple $(a,b,c)$ of positive integers is called
{\it quasi-Pythagorean\/} if there exists a triangle with
lengths of the sides $a$, $b$, $c$ and the
angle opposite to the side $c$ equal to $120^\circ$. Prove
that if $(a,b,c)$ is a \q\ then $c$ has a prime divisor
bigger than $5$.


\item%3 
Find all pairs of positive integers $x,\,y$ which satisfy the equation
$\;{2x^2+5y^2=11(xy-11)}$.


\item%4 
Let $P$ be a polynomial with integer coefficients.
Suppose that for $n=1,2,3,\ldots,1998$ the number $P(n)$ is a
three-digit positive integer. Prove that the polynomial $P$
has no integer roots.


\item%5 
Let $a$ be an odd digit and $b$ an even digit. Prove that for every
positive integer~$n$ there exists a positive integer, divisible by
$2^n$, whose decimal representation contains no digits other than
$a$~and~$b$.


\item%6 
Let $P$ be a polynomial of degree $6$ and let $a$, $b$ be real
numbers such that $0<a<b$. Suppose that $P(a)=P(-a)$,
$P(b)=P(-b)$, $P'(0)=0$. Prove that $P(x)=P(-x)$ for all real $x$.


\item%7 
Let $\R$ be the set of all real numbers.  Find all functions
${f\colon\R\to\R}$ satisfying for all ${x,y\in\R}$ the equation
$\;f(x)+f(y)=f(f(x)f(y)).$


\item%8  
Let ${P_k(x)=1+x+x^2+\cdots+x^{k-1}}$. Show that
$$\;\displaystyle{\sum_{k=1}^n{n\choose k}P_k(x)
=2^{n-1}P_n\Bigl({1+x\over2}\Bigr)}\;$$
for every real number $x$ and every positive integer~$n$.


\item%9 
Let the numbers $\alpha$, $\beta$ satisfy $0<\alpha<\beta<\pi/2$
and let $\gamma$ and $\delta$ be the numbers defined by the
conditions:\hfill\break (i) \ $0<\gamma<\pi/2$, and $\tan\gamma$
is the arithmetic mean of $\tan\alpha$ and
$\tan\beta$;\hfill\break (ii) \ $0<\delta<\pi/2$, and
$\displaystyle{1\over\cos\delta}$ is the arithmetic mean of
$\displaystyle{1\over\cos\alpha}$ and
$\displaystyle{1\over\cos\beta}$.\hfill\break Prove that
$\gamma<\delta$.


\item%10 
Let ${n\ge4}$ be an even integer. A~regular $n$-gon and a regular
${(n{-}1)}$-gon are inscribed into the unit circle. For each
vertex of the $n$-gon consider the distance from this vertex to
the nearest vertex of the ${(n{-}1)}$-gon, measured along the
circumference. Let $S$ be the sum of these $n$ distances. Prove
that $S$ depends only on~$n$, and not on the relative position of
the two polygons.


\item%11 
Let $a$, $b$, $c$ be the lengths of the sides of a triangle.
Let $R$ denote its circumradius. Prove that
$$R\geq{{a^{2}+b^{2}}\over {2\,\sqrt{2a^{2}+2b^{2}-c^{2}}}}\ .$$
When does equality hold$\,$?


\item%12 
In a triangle $ABC$, $\angle BAC=90^\circ$. Point $D$ lies
on the side $BC$ and satisfies $\angle BDA=2\angle BAD$.
Prove that
$$\frac{1}{AD}=\frac{1}{2}\left(\frac{1}{BD}+\frac{1}{CD}\right).$$


\item%13 
In a convex pentagon $ABCDE$, the sides $AE$ and $BC$ are
parallel and $\angle ADE=\angle BDC$. The diagonals $AC$
and $BE$ intersect in $P$. Prove that $\angle EAD=\angle
BDP$ and $\angle CBD=\angle ADP$.


\item%14 
Given triangle $ABC$ with $AB<AC$. The line passing through
$B$ and parallel to $AC$ meets the external angle bisector
of $\angle BAC$ at $D$. The line passing through $C$ and
parallel to $AB$ meets this bisector at $E$. Point $F$ lies
on the side $AC$ and satisfies the equality $FC=AB$. Prove
that $DF=FE$.


\item%15 
Given acute triangle $ABC$. Point $D$ is the foot of the
perpendicular from $A$ to $BC$. Point $E$ lies on the segment $AD$
and satisfies the equation $${AE\over ED}={CD\over DB}.$$ Point
$F$ is the foot of the perpendicular from $D$ to $BE$. Prove that
$\angle AFC=90^\circ$.


\item%16 
Is it possible to cover a $13\times 13$ chessboard with fourty-two
pieces of dimensions $4\times 1$ such that only the central field
of the chessboard remains uncovered$\,$? (We assume that each
piece covers exactly four fields of the chessboard.)


\item%17 
Let $n$ and $k$ be positive integers. There are $nk$ objects (of
the same size) and $k$ boxes, each of which can hold $n$ objects.
Each object is coloured in one of $k$ different colours. Show that
the objects can be packed in the boxes so that each box holds
objects of at most two colours.


\item%18 
Determine all positive integers $n$ for which there exists a set
$S$ with the following properties:\hfill\break (i) $S$ consists of
$n$ positive integers, all smaller than $2^{n-1}$; \hfill\break
(ii) for any two distinct subsets $A$ and $B$ of $S$, the sum of
the elements of $A$ is different from the sum of the elements of
$B$.



\item%19 
Consider a ping-pong match between two teams, each consisting of 1000
players. Each player played against each player of the other team
exactly once (there are no draws in ping-pong). Prove that there exist
ten players, all from the same team, such that every member of the
other team has lost his game against at least one of those ten players.


\item%20 
We say that some positive integer $m$ {\it covers\/} the
number 1998, if 1, 9, 9, 8 appear in this order as digits
of $m$. (For instance 1998 is covered by
2{\bf 1}59{\bf 9}36{\bf 98} but not
by 213326798.) Let $k(n)$ be the number of positive integers
that cover 1998 and have exactly $n$ digits ($n\geq 5$),
all different from $0$. What is the remainder of $k(n)$
in division by $8$?
\end{list}
\end{document}
