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\begin{center}
{\large\bf Problems for the Team Competition Baltic Way 1997}\\
Copenhagen, November 9, 1997 at 10 a.m.\\
$4\textstyle{\frac{1}{2}}$ hours.\quad 5 points per problem.
\end{center}

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

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\item
Determine all functions $f$ from the real
numbers to the real numbers, different from the zero
function, such that
$\,f(x)f(y)=f(x-y)\,$ 
for all real numbers $x$ and $y$.

\item
Given a sequence $a_1,a_2,a_3,\dots$ of positive integers in
which every positive integer occurs exactly once. Prove that
there exist integers $\ell$ and $m$, $1<\ell<m$,
such that $a_1+a_m=2a_{\ell}$.

\item
Let $x_1=1$ and
$x_{n+1}=x_n+\left\lfloor\frac{x_n}{n}\right\rfloor+2$, for
$n=1,2,3,\ldots$, 
where $\lfloor x\rfloor$ denotes the largest integer not
greater than $x$. Determine $x_{1997}$.

\item
Prove that the arithmetic mean $a$ of $x_1,\ldots,x_n$
satisfies
$$(x_1-a)^2+\cdots+(x_n-a)^2\le{1\over2}(|x_1-a|+\cdots+|x_n-a|)^2\,.$$

\item
In a sequence $u_0,u_1,\ldots$ of positive integers, $u_0$ is
arbitrary, and for any non-negative integer $n$,
$$u_{n+1}=\left\{\begin{array}{ll}
\textstyle\frac{1}{2}u_n&\mbox{for even $u_n$\,,}\\
a+u_n&\mbox{for odd $u_n$\,,}\end{array}\right.$$
where $a$ is a fixed odd positive integer.
Prove that the sequence is periodic from a certain step.

\item
Find all triples $(a,b,c)$ of non-negative integers
satisfying $a\ge b\ge c$ and
$$1{\cdot}a^3 + 9{\cdot}b^2 + 9{\cdot}c + 7 = 1997.$$

\item
Let $P$ and $Q$ be polynomials with integer coefficients. 
Suppose that the integers $a$ and $a+1997$ are roots of
$P$, and that $Q(1998)=2000$. Prove that the equation
$Q(P(x))=1$ has no integer solutions.

\item
If we add $1996$ to $1997$, we first add the unit digits $6$
and $7$. Obtaining $13$, we write down $3$ and ``carry'' $1$
to the next column.  Thus we make a {\it carry\/}.
Continuing, we see that we are to make three 
carries in total: \arraycolsep=1pt
$$\begin{array}{ccccc}
&\mbox{\scriptsize $1$}&\mbox{\scriptsize $1$}
&\mbox{\scriptsize $1$}&\\[-.5ex]
&1&9&9&6\\[-.5ex]
+\;&1&9&9&7\\
\hline
&3&9&9&3
\end{array}$$
Does there exist a positive integer $k$ such that 
adding $1996{\cdot}k$ to $1997{\cdot}k$ 
no carry arises during the whole calculation?

\item
The worlds in the Worlds' Sphere are numbered $1, 2, 3, \ldots$
and connected so that for any integer $n\geq 1$, Gandalf the
Wizard can move in both directions between any worlds with
numbers $n$, $2n$ and $3n+1$. Starting his travel from an
arbitrary world, can Gandalf reach every other world?

\item
Prove that in every sequence of 79 consecutive positive
integers written in the decimal system, there is a
positive integer whose sum of digits is divisible by 13.

\item
\begin{minipage}[t]{13cm}
On two parallel lines, the distinct points $A_1$,~$A_2$,
$A_3$, $\ldots$ respectively $B_1$,~$B_2$, $B_3$, $\dots$
are marked in such a way that $|A_iA_{i+1}|=1$ and
$|B_iB_{i+1}|=2$ for $i=1,2,\ldots$ (see figure).
Provided that $\angle A_1A_2B_1=\alpha$, find the infinite
sum $\;\angle A_1B_1A_2 + \angle A_2B_2A_3 + \angle A_3B_3A_4
+ \cdots.$
\end{minipage}
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\item
Two circles ${\cal C}_1$ and ${\cal C}_2$ intersect in $P$
and $Q$. A line through $P$ intersects ${\cal C}_1$ and
${\cal C}_2$ again in $A$ and $B$, respectively, and $X$ is
the midpoint of $AB$. The line through $Q$ and $X$ intersects
${\cal C}_1$ and ${\cal C}_2$ again in $Y$ and $Z$,
respectively. Prove that $X$ is the midpoint of $YZ$.

\item
\begin{minipage}[t]{13cm}
Five distinct points $A$,~$B$, $C$, $D$
and~$E$ lie on a line with $|AB|=|BC|=|CD|=|DE|$. 
The point~$F$ lies outside the line. Let $G$
be the circumcentre of the triangle~$ADF$ and $H$  the
circumcentre of the triangle~$BEF$. Show that the
lines~$GH$ and~$FC$ are perpendicular.
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\item
In the triangle $ABC$, $|AC|^2$ is the arithmetic mean of
$|BC|^2$ and $|AB|^2$. Show that $\cot^2 B\ge\cot A\cdot\cot C$.

%\pr{15}{g12}
\item
In the acute triangle $ABC$, the bisectors of $\angle A$,
$\angle B$ and $\angle C$ intersect the circumcircle again
in $A_1$, $B_1$ and $C_1$, respectively. Let $M$ be the
point of intersection of $AB$ and $B_1C_1$, and let $N$ be
the point of intersection of $BC$ and $A_1B_1$. Prove that
$MN$ passes through the incentre of $\triangle ABC$.

%\pr{16}{d4}
\item
On a $5\times 5$ chessboard, two players play the following
game: The first player places a knight on some square. Then
the players alternately move the knight according to the
rules of chess, starting with the second player. It is not
allowed to move the knight to a square that was visited
previously. The player who cannot move loses.  Which of the
two players has a winning strategy?

%\pr{17}{d5}
\item
A rectangle can be divided into $n$ equal squares. The same
rectangle can also be divided into $n+76$ equal squares.
Find $n$. 

%\pr{18}{d9}
\item
({\it i\/}) Prove the existence of two infinite sets $A$ and
$B$, not necessarily disjoint, of non-negative integers
such that each non-negative 
integer $n$ is uniquely representable in the form $n=a+b$
with $a\in A$, $b\in B$.\\
({\it ii\/}) Prove that for each such pair
$(A,B)$, either $A$ or $B$ contains only multiples of some
integer $k>1$.

%\pr{19}{d6}
\item
In a forest each of $n$ animals $(n\geq3)$ lives in its own
cave, and there is exactly one separate path between any two
of these caves.  Before the election for King of the Forest
some of the animals make an election campaign.  Each
campaign-making animal visits each of the other caves
exactly once, uses only the paths for moving from cave to
cave, never turns from one path to another between the caves
and returns to its own cave in the end of its campaign. It
is also known that no path between two caves is used by more
than one campaign-making animal.\par
\begin{itemize}
\item[{\it a\/})]
  Prove that for any prime $n$, the maximum possible number of
  campaign-making animals is $\frac{n-1}{2}$;
\item[{\it b\/})]
  Find the maximum number of campaign-making animals for $n=9$.
\end{itemize}

%\pr{20}{d7}
\item
Twelve cards lie in a row. The cards are of three kinds: with both
sides white, both sides black, or with a white and a black side.
Initially, nine of the twelve cards have a black side up. The
cards~1--6 are turned, and subsequently four of the twelve cards have
a black side up. Now cards~4--9 are turned, and six cards have a black
side up. Finally, the cards 1--3 and~10--12 are turned, after which
five cards have a black side up. How many cards of each kind were
there?

\end{list}

\end{document}

