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{\large\bf Problems for the Team Competition Baltic Way 1999}\\
Reykjavik, November 6, 1999
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%\magnification1200
\font\tlustazupa=cmbx10 scaled\magstep2
\font\zupa=cmr10 scaled\magstep1
\font\upa=msbm10 scaled\magstep0
\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
Determine all real numbers $a$, $b$, $c$, $d$ that satisfy the following equations
\arraycolsep=1pt
$$\left\{\begin{array}{rcl}
abc+ab+bc+ca+a+b+c&=&1\\
bcd+bc+cd+db+b+c+d&=&9\\
cda+cd+da+ac+c+d+a&=&9\\
dab+da+ab+bd+d+a+b&=&9\,.
\end{array}\right. $$

\item%2 
Determine all positive integers $n$ with the property that the third root of
$n$ is obtained by removing its last three decimal digits.

\item%3 
Determine all positive integers $n\geq 3$ such that the inequality
$$a_1a_2+a_2a_3+\ldots+a_{n-1}a_n+a_na_1\leq0$$
holds for all real numbers $a_1,a_2,\ldots,a_n$ which satisfy
$a_1+a_2+\ldots+a_n=0\,.$

\item%4 
For all positive real numbers $x$ and $y$ let
$$f(x,y)=\min\left(x,\frac{y}{x^2+y^2}\right)\,.$$
Show that there exist $x_0$ and $y_0$ such that
$f(x,y)\leq f(x_0,y_0)$ for all positive $x$ and $y$,
and find $f(x_0,y_0)$.

\item%5 
The point $(a,b)$ lies on the circle $x^2+y^2=1$. The tangent to the circle
at this point meets the parabola $y=x^2+1$ at exactly one point. 
Find all such points $(a,b)$.

\item%6 
What is the least number of moves it takes a knight to get from
one corner of an $n\times n$ chessboard, where $n\geq4$, to the diagonally
opposite corner?

\item%7 
Two squares on an $8\times8$ chessboard are called adjacent if they have 
a common edge or common corner. Is it possible for a king to begin in some square and
visit all squares exactly once in such a way that all moves except the first are
made into squares adjacent to an even number of squares already visited?

\item%8  
We are given 1999 coins. No two coins have the same weight. A machine is
provided which allows us with one operation to determine, for any three
coins, which one has the middle weight. Prove that the coin that is the
$1000$th by weight can be determined using no more than $1000000$ operations
and that this is the only coin whose position by weight can be determined
using this machine. 

\item%9 
A cube with edge length $3$ is divided into $27$ unit cubes. The
numbers $1,2,\ldots,27$ are distributed arbitrarily over the unit 
cubes, with one number in each cube. We form the $27$ possible
row sums (there are nine such sums of three integers for each of 
the three directions parallel with the edges of the cube). 
At most how many of the 27 row sums can be odd?

\item%10 
May the points of a disc of radius $1$ (including its circumference)
be partitioned into three subsets in such a way that no subset contains two
points separated by a distance $1$?

\item%11 
Prove that for any four points in the plane, no three of which are collinear,
there exists a circle such that three of the four points are on the circumference
and the fourth point is either on the circumference or inside the circle.

\item%12 
In a triangle $ABC$ it is given that $2AB=AC+BC$. Prove that the incentre
of $ABC$, the circumcenter of $ABC$, and the midpoints of $AC$ and $BC$ are
concyclic.

\item%13 
The bisectors of the angles $A$ and $B$ of the triangle $ABC$ meet the sides
$BC$ and $CA$ at the points $D$ and $E$, respectively. Assuming that $AE+BD=AB$,
determine the angle $C$.


\item%14 
Let $ABC$ be an isosceles triangle with $AB=AC$. Points $D$ and $E$ lie on the
sides $AB$ and $AC$, respectively. The line passing through $B$ and parallel to
$AC$ meets the line $DE$ at $F$. The line passing through $C$ and parallel to
$AB$ meets the line $DE$ at $G$. Prove that
$$\frac{[DBCG]}{[FBCE]}=\frac{AD}{AE}\,,$$
where $[PQRS]$ denotes the area of the quadrilateral $PQRS$.

\item%15 
Let $ABC$ be a triangle  with $\angle C=60^\circ$ and $AC<BC$.
The point $D$ lies on the side $BC$ and satisfies $BD=AC$.
The side $AC$ is extended to the point $E$ where  $AC=CE$.
Prove that $AB=DE$.


\item%16 
Find the smallest positive integer $k$ which is representable in the form
$k=19^n-5^m$ for some positive integers $m$ and $n$.

\item%17 
Does there exist a finite sequence of integers $c_1,c_2,\ldots,c_n$
such that all the numbers  $a+c_1$, $a+c_2$, $\ldots$,
$a+c_n$ are primes for more than one but not infinitely many
different integers $a$?

\item%18 
Let $m$ be a positive integer such that $m\equiv 2 (\mbox{mod}\;4)$.
Show that there exists at most one factorization $m=ab$ where $a$ and $b$
are positive integers satisfying $$0<a-b<\sqrt{5+4\sqrt{4m+1}}\,.$$

\item%19 
Prove that there exist infinitely many even positive integers $k$ such that 
for every prime $p$ the number $p^2+k$ is composite.

\item%20 
Let $a$, $b$, $c$ and $d$ be prime numbers such that ${a>3b>6c>12d}$ and
$a^2-b^2+c^2-d^2=1749$. Determine all possible values of $a^2+b^2+c^2+d^2\,.$
\end{list}
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