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\title{Baltic Way 2000}
\author{}\date{Oslo, November 5, 2000}
\maketitle

\noindent

\subsubsection*{Problem 1}
Let $K$ be a point inside the triangle $ABC$. Let $M$ and $N$ be
points such that $M$ and $K$ are on opposite sides of the line $AB$,
and $N$ and $K$ are on opposite sides of the line $BC$.
Assume that $\angle MAB=\angle MBA=\angle
NBC=\angle NCB=\angle KAC=\angle KCA$. Show that $MBNK$ is a
parallelogram. 


\subsubsection*{Problem 2}
Given an isosceles triangle $ABC$
with $\angle A=90^\circ$. Let $M$ be the midpoint of $AB$.
The line passing through $A$ and perpendicular to $CM$
intersects the side $BC$ at $P$. Prove that 
$$\angle AMC=\angle BMP.$$


\subsubsection*{Problem 3}
Given a triangle $ABC$
with $\angle A=90^\circ$ and $AB\neq AC$. The points $D$,
$E$, $F$ lie on the sides $BC$, $CA$, $AB$, respectively,
in such a way that $AFDE$ is a square. Prove that
the line $BC$, the line $FE$ and the line tangent at 
the point $A$ to the circumcircle of the triangle $ABC$
intersect in one point.


\subsubsection*{Problem 4}
Given a triangle $ABC$ with $\angle A=120^\circ$.
The points $K$ and $L$ lie on the sides $AB$ and $AC$,
respectively.
Let $BKP$ and $CLQ$ be equilateral triangles
constructed outside the triangle $ABC$. Prove that
$\;PQ\geq \frac{\sqrt{3}}{2}\,(AB+AC)\,.$


\subsubsection*{Problem 5} Let $ABC$ be a triangle such that 
$${BC\over AB-BC}={AB+BC\over AC}.$$
Determine the ratio $\angle A:\angle C$.


\subsubsection*{Problem 6}
Fredek runs a private hotel. He claims that whenever
$n\ge3$ guests visit the hotel, it is possible to select two guests that
have equally many acquaintances among the other guests, and that also have a
common acquaintance or a common unknown among the guests.
For which values of $n$ is Fredek right?

\noindent
(Acquaintance is a symmetric relation.)

\subsubsection*{Problem 7}
In a $40 \times 50$ array of control buttons, each button has two states:
on and off.  By touching a button, its state and the states of all
buttons in the same row and in the same column are switched. Prove that the array of
control buttons may be altered from the all-off state to the
all-on state by touching buttons successively, and determine the
least number of touches needed to do so.


\subsubsection*{Problem 8}
Fourteen friends met at a party.  One of them, Fredek, wanted to go
to bed early.  He said goodbye to 10 of his friends, forgot about the
remaining 3, and went to bed.  After a while he returned to the party,
said goodbye to 10 of his friends (not necessarily the same as before),
and went to bed.  Later Fredek came back a number of times, each time
saying goodbye to exactly 10 of his friends, and then went back to bed.
As soon as he had said goodbye to each of his friends at least once,
he did not come back again.  In the morning Fredek realised that he had
said goodbye a different number of times to each of his thirteen friends!
What is the smallest possible number of times that Fredek returned to
the party?


\subsubsection*{Problem 9} There is a frog jumping on a $2k\times 2k$
chessboard, composed of unit squares. The frog's jumps are
$\sqrt{1+k^2}$ long and
they carry the frog from the center of a square to the center of
another
square. Some $m$ squares of the board are marked with an $\times$, and
all the squares into which the frog can jump from an $\times$'d square 
(whether they carry an $\times$ or not) are marked with an $\circ$. 
There are $n$ $\circ$'d squares. Prove that $n\ge m$.


\subsubsection*{Problem 10}
Two positive integers are written on the blackboard.
Initially, one of them is 2000 and the other is smaller than 2000. 
If the arithmetic mean $m$ of the two numbers on the blackboard is an
integer, the following operation is allowed: 
One of the two numbers is erased and replaced by $m$.
Prove that this operation cannot be performed more than ten times. Give an
example where the operation is performed ten times.


\subsubsection*{Problem 11}
A sequence of positive integers $a_1,a_2,\dots$ is such that for each
$m$
and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$,
then
$a_m$ is a divisor of $a_n$ and $a_m<a_n$. Find the least possible
value of $a_{2000}$.


\subsubsection*{Problem 12}
Let $x_1, x_2,\ldots, x_n$ be positive integers such that no 
one of them is an initial fragment of any other (for example, 
12 is an initial fragment of $\underline{12}$, $\underline{12}5$ and 
$\underline{12}405$).
Prove that $$\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}<3.$$


\subsubsection*{Problem 13}
Let $a_1,a_2,\dots,a_n$ be an arithmetic progression of 
integers such that
$i\,|\,a_i$ for $i=1,2,\dots,n-1$ and $n\not| a_n$. Prove 
that $n$ is a prime power.


\subsubsection*{Problem 14}
Find all positive integers $n$ such that $n$ is equal to $100$
times the number of positive divisors of $n$.


\subsubsection*{Problem 15}
Let $n$ be a positive integer not divisible by 2 or 3. 
Prove that for all integers $k$, the number
$(k+1)^n-k^n-1$ is divisible by $k^2+k+1$.


\subsubsection*{Problem 16}
Prove that for all positive real numbers $a$, $b$, $c$ we have
$$\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}\geq\sqrt{a^2+ac+c^2}.$$


\subsubsection*{Problem 17}
Find all real solutions to the following system of
equations:
\begin{displaymath}
     \left\{ \begin{array}{ccccccccc}
     x &+ &y &+ &z &+ &t &= &5\\
     xy &+ &yz &+ &zt &+ &tx &= &4\\
     xyz &+ &yzt &+ &ztx &+ &txy &= &3\\
     xyzt &  &   &   &    &  &    &= &-1.\end{array}
     \right.
\end{displaymath}


\subsubsection*{Problem 18}
Determine all positive real numbers $x$ and $y$ satisfying the equation
$$x+y+\frac 1x+\frac 1y+4=2\cdot(\sqrt{2x+1}+\sqrt{2y+1}).$$


\subsubsection*{Problem 19} Let $t\geq\frac12$ be a
real number and $n$ a positive integer. 
Prove that $$t^{2n}\geq (t-1)^{2n}+(2t-1)^n.$$


\subsubsection*{Problem 20}
For every positive integer $n$, let
$$x_n=\frac{(2n+1)(2n+3)\cdots(4n-1)(4n+1)}
{(2n)(2n+2)\cdots(4n-2)(4n)}.$$
Prove that $\frac{1}{4n}<x_n-\sqrt2<\frac2n$.




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