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%    BALTIC WAY 2001 - PROBLEMS                                                                       %
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%  last change on:  Nov 3, 2001                                                                      %
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\begin{document}

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\centering\subsection*{Problems for the Team Competition Baltic Way 2001}
Hamburg, November 4, 2001

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\begin{list_1}

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%    COMBINATORICS                                                                                    %
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\item   %   C 4   (LAT 3)
A set of 8 problems was prepared for an examination. Each
student was given 3 of them. No two students received more than one common
problem. What is the largest possible number of students?

\item   %   C 5   (LAT 6)
Let $n\ge 2$ be a positive integer. Find whether there exist
$n$ pairwise nonintersecting nonempty subsets of $\{1,2,3,\dots\}$
such that each positive integer can be expressed in a unique way
as a sum of at most $n$ integers, all from different subsets.

\item   %   C 6   (SWE 3)
The numbers $1, 2, \ldots , 49$ are placed in a $7 \times 7$
array, and the sum of the numbers in each row and in each column is
computed. Some of these $14$ sums are odd while others are even. Let
$A$ denote the sum of all the odd sums and $B$ the sum of all even sums.
Is it possible that the numbers were placed in the array in such a way
that $A = B$?

\item   %   C 10   (DEN 1)
Let $p$ and $q$ be two different primes. Prove that
\[ \left\lfloor\frac{p}{q}\right\rfloor
   + \left\lfloor\frac{2p}{q}\right\rfloor
   + \left\lfloor\frac{3p}{q}\right\rfloor
   + \cdots + \left\lfloor\frac{(q-1)p}{q}\right\rfloor
   = \frac{1}{2}(p-1)(q-1) \quad. \]
(Here $\lfloor x\rfloor$ denotes the largest integer not greater than $x$.)

\item   %   C 12   (GER 2)
Let 2001 given points on a circle be colored either red or
green. In one step all points are recolored simultaneously in
the following way: If both direct neighbors of a point $P$ have
the same color as $P$, then the color of $P$ remains unchanged,
otherwise $P$ obtains the other color. Starting with the first
coloring $F_1$, we obtain the
colorings $F_2, F_3,\ldots$ after several recoloring steps.
Prove that there is a number $n_0\leq
1000$ such that $F_{n_0}=F_{n_0+2}$. Is the assertion also true if
1000 is replaced by 999?

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%    GEOMETRY                                                                                         %
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\item   %   G 1   (FIN 1)
The points $A,B,C,D,E$ lie on the circle $c$ in this order and satisfy
$AB\parallel EC$ and $AC\parallel ED$. The line tangent to the circle $c$ at $E$
meets the line $AB$ at $P$. The lines $BD$ and $EC$ meet at $Q$.
Prove that $|AC|=|PQ|$.

\item   %   G 2   (LAT 4)
Given a parallelogram $ABCD$. A circle passing through $A$ meets the line
segments
$AB$, $AC$ and $AD$ at inner points $M$, $K$, $N$, respectively.
Prove that
$$ |AB|\cdot|AM|+|AD|\cdot|AN|=|AK|\cdot |AC|. $$

\item   %   G 3   (LAT 7)
Let $ABCD$ be a convex quadrilateral, and let $N$ be
the midpoint of $BC$. Suppose further that $\angle AND= 135^\circ$.
Prove that
$\ds |AB|+|CD|+\frac{1}{\sqrt{2}}\cdot|BC|\ge|AD|.$

\item   %   G 6   (POL 3)
Given a rhombus $ABCD$, find the locus of the
points $P$ lying inside the rhombus and satisfying
$\angle APD+\angle BPC=180^\circ$.

\item   %   G 8   (POL 5)
In a triangle $ABC$, the bisector of
$\angle BAC$ meets the side $BC$ at the point $D$.
Knowing that $|BD|\cdot |CD|=|AD|^2$ and $\angle ADB=45^\circ$,
determine the angles of triangle $ABC$.

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%    ALGEBRA                                                                                          %
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\item   %   A 2   (FIN 4)
The real-valued function $f$ is defined for all positive
integers. For any integers $a&gt;1$, $b&gt;1$ with $d=\gcd(a,\,b)$, we have
$$f(ab)=f(d)\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right).$$
Determine all possible values of $f(2001)$.

\item   %   A 3   (ICE 3)
Let $a_{1}, a_{2}, ... , a_{n}$ be positive real numbers such that
$\ds\sum_{i=1}^{n} a_{i}^{3}=3$
and $\ds\sum_{i=1}^{n} a_{i}^{5}=5$.
Prove that $\ds\sum_{i=1}^{n} a_{i}&gt;3/2$.

\item   %   A 4   (LAT 1)
Let $a_0,a_1,a_2,\dots$ be a sequence of real numbers satisfying
$a_0=1$ and
$\ds a_n=a_{\left\lfloor 7n/9\right\rfloor}
         +a_{\left\lfloor n/9\right\rfloor}$
for $n=1,2,\dots$.
Prove that there exists a positive integer $k$ with $\ds a_k&lt;\frac{k}{2001!}$.\\
(Here $\lfloor x\rfloor$ denotes the largest integer not greater
than $x$.)

\item   %   A 5   (LAT 5)
There are $2n$ cards. On each card some real number $x$, $1\le x\le 2$, is
written (there can be different numbers on different cards).
Prove that the cards can be divided into two heaps with sums
$s_1$ and $s_2$ so that $\ds\frac{n}{n+1}\le\frac{s_1}{s_2}\le 1$.

\item   %   A 8   (GER 1)
Let $a_0,a_1,a_2,\dots$ be a sequence of positive real
numbers satisfying
$i\cdot a_i^2\ge (i+1)\cdot a_{i-1}a_{i+1}$ for $i=1,2,\dots$
Furthermore, let $x$ and $y$ be positive
reals, and let $\ds b_i=xa_i+ya_{i-1}$ for $i=1,2,\dots$.
Prove that the inequality
$i\cdot b_i^2&gt;(i+1)\cdot b_{i-1}b_{i+1}$
holds for all integers $i\ge 2$.

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%    NUMBER THEORY                                                                                    %
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\item   %   N 1   (ICE 1)
Let $f$ be a real-valued function defined
on the positive integers satisfying the following condition: For
all $n&gt;1$ there exists a prime divisor $p$ of $n$ such that
$f(n)=f(n/p)-f(p)$. Given that $f(2001)=1$, what is the value
of $f(2002)$?

\item   %   N 3   (LAT 2)
Let $n$ be a positive integer.
Prove that at least $2^{n-1}+n$ numbers can be chosen from
the set $\{1,2,3,\dots,2^n\}$ such that for any two different
chosen numbers $x$ and $y$, $x+y$ is not a divisor of $x\cdot y$.

\item   %   N 6   (DEN 2)
Let $a$ be an odd integer. Prove that $a^{2^n} +
2^{2^n}$ and $a^{2^m} + 2^{2^m}$ are relatively prime for all
positive integers $n$ and $m$ with $n \neq m$.

\item   %   N 9   (SWE 2)
What is the smallest positive odd integer having the same number
of positive divisors as 360?

\item   %   N 10   (LAT 8)
From a sequence of integers $(a, b, c, d)$ each of the sequences
$$(c,d,a,b),\quad (b,a,d,c),\quad (a+nc,b+nd,c,d),\quad (a+nb,b,c+nd,d)$$
for arbitrary integer $n$ can be obtained by one step.
Is it possible to obtain $(3, 4, 5, 7)$ from $(1, 2, 3, 4)$ through a
sequence of such steps?

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\end{document}
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