\nopagenumbers
\input amssym.def
\input amssym.tex
\def\geq{\geqslant}
\def\leq{\leqslant}
\centerline{\bf M A T H E M A T I C S}
\centerline{\bf IX International Olympiad ``Tuymaada---2002"}
\centerline{(Senior league)}
\bigskip
\centerline{\bf Day 1}
\bigskip
{\bf 1.}\hskip 0.3cm
Each of the points $G$ and $H$ lying from different sides of the plane
of hexagon $ABCDEF$ is connected with all vertices of the
hexagon. Is it possible to mark 18 segments thus formed
by the numbers 1, 2, 3, \dots, 18 and arrange some real numbers
at points $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ so that each segment
is marked with the difference of the numbers at its ends?
\rightline{(A.Golovanov)}
\smallskip
{\bf 2.}\hskip 0.3cm
The product of positive numbers $a$, $b$, $c$, and $d$ is 1. Prove that
$${1+ab\over 1+a}+{1+bc\over 1+b}+{1+cd\over 1+c}+{1+da\over 1+d}\geq 4.$$
\rightline{(A.Khrabrov)}
\smallskip
{\bf 3.}\hskip 0.3cm
A circle having common centre with the circumcircle of triangle $ABC$
meets the sides of the triangle at six points forming convex
hexagon $A_1A_2B_1B_2C_1C_2$ ($A_1$ and $A_2$ lie on $BC$, $B_1$ and $B_2$
lie on $AC$, $C_1$ and $C_2$ lie on $AB$).
If $A_1B_1$ is parallel to the bisector of angle $B$,
prove that $A_2C_2$ is parallel to the bisector of angle $C$.
\rightline{(S.Berlov)}
\smallskip
{\bf 4.}\hskip 0.3cm
A rectangular table with 2001 rows and 2002 columns is partitioned
into $1\times 2$ rectangles. It is known that any other partition
of the table into $1\times 2$ rectangles contains a rectangle belonging
to the original partition. Prove that the original partition contains
two successive columns covered by 2001 horizontal rectangles.
\rightline{(S.Volchenkov)}
\bigskip
\centerline{\bf M A T H E M A T I C S}
\centerline{\bf IX International Olympiad "Tuymaada---2002"}
\centerline{(Senior league)}
\bigskip
\centerline{\bf Day 2}
\bigskip
{\bf 5.}\hskip 0.3cm
A positive integer $c$ is given. The sequence $\{p_k\}$ is constructed
by the following rule: $p_1$ is arbitrary prime and for $k\geq 1$
the number $p_{k+1}$ is any prime divisor of $p_k+c$ not present among
the numbers $p_1$, $p_2$, \dots, $p_k$. Prove that the sequence
$\{p_k\}$ cannot be infinite.
\rightline{(A.Golovanov)}
\smallskip
{\bf 6.}\hskip 0.3cm
Find all the functions $f(x)$, continuous on the whole real axis, such
that for every real $x$
$$f(3x-2)\leq f(x)\leq f(2x-1).$$
\rightline{(A.Golovanov)}
\smallskip
{\bf 7.}\hskip 0.3cm
The points $D$ and $E$ on the circumcircle of an acute triangle $ABC$ are
such that $AD=AE$. Let $H$ be the common point of the altitudes of triangle
$ABC$. It is known that $AH^2=BH^2+CH^2$. Prove that $H$ lies on the segment
$DE$.
\rightline{(D.Shiryaev)}
\smallskip
{\bf 8.}\hskip 0.3cm
A real number $\alpha$ is given. The sequence $n_1