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\input amssym.def
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\def\geq{\geqslant}
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\centerline{\bf ``TUYMAADA--2004''}
\medskip
\centerline{\it Senior league}
\medskip
\centerline{\tt First day}
\medskip
\q1. Do there exist a sequence $a_1$, $a_2$, $a_3$, \dots of real numbers
and a non-constant polynomial $P(x)$ such that $a_m+a_n=P(mn)$ for
every positive integral $m$ and $n$?
\rightline{({\it A. Golovanov}\/)}
\q2. 100 straight lines are drawn in the plane so that no two of them
are parallel and no three of them meet at one point. All the points
of their intersection are marked. Then all the lines and $k$ of the marked
points are erased. Given the remained points of intersection, for
what maximum $k$ one can always reconstruct the
original lines?
\rightline{({\it A. Golovanov}\/)}
\q3. An acute triangle $ABC$ is inscribed in a circle of radius 1
with centre $O$; all the angles of $ABC$ are greater than $45^\circ$.
$B_1$ is the foot of perpendicular from $B$ to $CO$,
$B_2$ is the foot of perpendicular from $B_1$ to $AC$.
Similarly, $C_1$ is the foot of perpendicular from $C$ to $BO$,
$C_2$ is the foot of perpendicular from $C_1$ to $AB$. The lines
$B_1B_2$ and $C_1C_2$ intersect at $A_3$. The points $B_3$ and $C_3$
are defined in the same way.
Find the circumradius of triangle $A_3B_3C_3$.
\rightline{({\it F.Bakharev, F.Petrov}\/)}
\q4. There are many opposition societies in the city of N.
Each society consists of $10$ members. It is known that for
every 2004 societies there is a person belonging at least to 11 of them.
Prove that the government can arrest 2003 people so that at least
one member of each society is arrested.
\rightline{({\it V.Dolnikov, D.Karpov}\/)}
\medskip
\centerline{\tt Second day}
\medskip
\q5. 50 knights of King Arthur sat at the Round Table. A glass of
white or red wine stood before each of them. It is known that at least
one glass of red wine and at least one glass of white wine stood on the table.
The king clapped his hands twice. After the first clap every knight
with a glass of red wine before him took a glass from his left neighbour.
After the second clap every knight with a glass of white
wine (and possibly something more) before him gave this glass
to the left neughbour of his left neighbour. Prove that
some knight was left without wine.
\rightline{({\it A. Khrabrov, incorrect translation from Hungarian}\/)}
\q6. The incircle of triangle $ABC$ touches its sides $AB$ and $BC$
at points $P$ and $Q$. The line $PQ$ meets the circumcircle of triangle
$ABC$ at points $X$ and $Y$. Find $\angle XBY$ if $\angle ABC = 90^\circ$.
\rightline{({\it A. Smirnov}\/)}
\q7. Zeroes and ones are arranged in all the squares of $n\times n$ table.
All the squares of the left column are filled by ones, and the sum
of numbers in every figure of the form
\leavevmode\vbox{\lines2
_ _
|_|_|
|_|}
(consisting of a square and its neighbours from left and from below)
is even. Prove that no two rows of the table are identical.
\rightline{({\it O. Vanyushina}\/)}
\q8. It is known that $m$ and $n$ are positive integers, $m > n^{n-1}$, and
all the numbers $m+1$, $m+2$, \dots, $m+n$ are composite. Prove that
there exist such different primes $p_1$, $p_2$, \dots, $p_n$ that
$p_k$ divides $m+k$ for $k = 1$, 2, \dots, $n$.
\rightline{({\it C. A. Grimm}\/)}
\end