\nopagenumbers
\def\q#1. {{\bf #1. }}
\centerline{\bf ``TUYMAADA-2001''}
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\centerline{\it Senior league}
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\centerline{\tt First day}
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\q1. Ten volleyball teams played a tournament; every two teams met exactly
once. The winner of a game gets 1 point, the loser gets 0 (there
are no draws in volleyball). If the team that scored $n$-th has $x_n$
points ($n=1, \dots, 10$), prove that $x_1+2x_2+\dots+10x_{10}\ge 165$.
\rightline{({\it D. Teryoshin}\/)}
\q2. Solve the equation
$$
(a^2,b^2)+(a,bc)+(b,ac)+(c,ab)=199.
$$
in positive integers.
(Here $(x,y)$ denotes the greatest common divisor of $x$ and $y$.)
\rightline{({\it S. Berlov}\/)}
\q3. Do there exist quadratic trinomials $P$, $Q$, $R$ such that for
every integers $x$ and $y$ an integer $z$ exists satusfying $P(x)+Q(y)=R(z)$?
\rightline{({\it A. Golovanov}\/)}
\q4. Unit square $ABCD$ is divided into $10^{12}$ smaller squares
(not necessarily equal). Prove that the sum of perimeters of all
the smaller squares having common points with diagonal $AC$ does
not exceed 1500.
\rightline{({\it A. Kanel-Belov}\/)}
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\centerline{\tt Second day}
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\q5. All positive integers are distributed among two disjoint sets $N_1$ %???
and $N_2$ such that no difference of two numbers belonging to the same
set is a prime greater than 100.
Find all such distributions.
\rightline{({\it N. Sedrakyan}\/)}
\q6. Non-zero numbers are arranged in $n \times n$ square ($n>2$). Every
number is exactly $k$ times less than the sum of all the other numbers
in the same cross (i.e., $2n-2$ numbers written in the same row
or column with this number).
Find all possible $k$.
\rightline{({\it D. Rostovsky, A. Khrabrov, S. Berlov}\/)}
\q7. $ABCD$ is a convex quadrilateral; half-lines $DA$ and $CB$ meet at point
$Q$; half-lines $BA$ and $CD$ meet at point $P$. It is known that
$\angle AQB=\angle APD$. The bisector of angle $\angle AQB$ meets the
sides $AB$ and $CD$ of the quadrilateral at points $X$ and $Y$, respectively;
the bisector of angle $\angle APD$ meets the
sides $AD$ and $BC$ at points $Z$ and $T$, respectively.
The circumcircles of triangles $ZQT$ and $XPY$ meet at point $K$ inside
the quadrilateral.
Prove that $K$ lies on the diagonal $AC$.
\rightline{({\it S. Berlov}\/)}
\q8. Is it possible to colour all positive real numbers by 10 colours
so that every two numbers with decimal representations differing
in one place only are of different colours? (We suppose that
there is no place in a decimal representation such that all digits
starting from that place are 9's.)
\rightline{({\it A. Golovanov}\/)}
\end