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\input amssym.def
\input amssym.tex
\def\geq{\geqslant}
\def\leq{\leqslant}
\centerline{\bf ``TUYMAADA--2005''}
\medskip
\centerline{\it Senior league}
\medskip
\centerline{\tt First day}
\medskip
\q1. All positive integers 1, 2, \dots, 121 are arranged in the squares
of $11\times 11$ table. Dima found the product of numbers in each row;
Sasha found the product of numbers in each column. Could they get the same
set of 11 numbers?
\rightline{({\it S. Berlov}\/)}
\q2. Six members of the team of Fatalia for the International
Mathematical Olympiad are selected from 13 candidates.
At the Team
Selection Test the candidates got $a_1$, $a_2$, \dots, $a_{13}$ points
($a_i\ne a_j$ for $i\ne j$).
The Team Leader has already chosen 6 candidates and now wants to see
them and nobody other in the team. With that end in view he constructs
a polynomial $P(x)$ and finds the {\it creative potential} of each
candidate by the formula $c_i=P(a_i)$. For what minimum $n$ can he always
find a polynomial $P(x)$ of degree not exceeding $n$ such that
the creative potential of all his six candidates is strictly more than
that of the seven others?
\rightline{({\it F. Petrov, K. Sukhov}\/)}
\q3. Organizers of a mathematical congress found that if they
accomodate any participant in a single room the rest can be accomodated
in double rooms so that two persons living in every double room know each
other.
Prove that every participant can organize a round table on graph theory
for himself and even number of other people so that each participant of
the round table knows both his neighbours.
\rightline{({\it S. Berlov, S. Ivanov}\/)}
\q4. In a triangle $ABC$, $A_1$, $B_1$, and $C_1$ are the points where
excircles touch the sides $BC$, $CA$, and $AB$ respectively. Prove that
$AA_1$, $BB_1$, and $CC_1$ are sides of a triangle.
\rightline{({\it L. Emelyanov}\/)}
\medskip
\centerline{\tt Second day}
\medskip
\q5.
Several rooks stand in the squares of the table shown in the figure.
The rooks beat all the squares (we suppose that a rook beats the square
it stands in). Prove that one can remove several rooks so that not more
than 11 rooks are left and these rooks still beat all the squares.
\rightline{({\it D. Rostovsky, based on folklore}\/)}
\q6. Given are positive integer $n$ and infinite sequence of proper
fractions
$$ x_0 = {a_0 \over n}, \quad x_1 = {a_1 \over n+1}, \quad
x_2 = {a_2 \over n+2}, \quad \dots \qquad a_i 3 .$$
\rightline{({\it A. Khrabrov}\/)}
\end