\nopagenumbers
\input amssym.def
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\centerline{\bf ``TUYMAADA--2011''}
\medskip
\centerline{\it Senior league}
\medskip
\centerline{\tt First day}
\medskip
1. Red, blue, and green children stand in a circle. When a teacher asked
the red children that have a green neighbour to lift a hand, 20 children
lifted their hands. When she asked the blue children that have a green
neighbour to lift a hand, 25 children lifted their hands. Prove that
some child that lifted the hand had two green neighbours.
\rightline{({\it A. Golovanov}\/)}
2. Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$;
$M$ is the midpoint of $AB$. Points $S_1$ and $S_2$ lie on the line $AB$.
The tangents drawn from $S_1$ to $\omega_1$ touch it at $X_1$ and $Y_1$;
the tangents drawn from $S_2$ to $\omega_2$ touch it at $X_2$ and $Y_2$.
Prove that if the line $X_1X_2$ passes through $M$, then $Y_1Y_2$ also
passes through $M$.
\rightline{({\it A. Akopyan}\/)}
3. At each square of an infinite chessboard the minimum number
of moves allowing a knight to reach this square from a given square $O$
is written.
A square is called {\it singular} if 100 is written in it and 101 is written
in all the squares sharing a side with it. How many singular squares are there?
\rightline{({\it A. Golovanov}\/)}
4. There are exactly 10 biquadrates and 100 cubes among several
consecutive positive integers. Prove that there are at least 2000 perfect squares
among these positive integers.
\rightline{({\it A. Golovanov}\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. All the numbers greater than 1 are coloured in two colours (both colours are used). Prove that
there exist real $a$ and $b$ such that the numbers $a+{1\over b}$ and
$b+{1\over a}$ have different colours.
\rightline{({\it A. Golovanov}\/)}
6. In a word of more than 10 letters, every two consecutive letters
are different. Prove that one can change places of two consecutive letters
so that the resulting word is not periodic (that is, cannot be divided
into equal subwords).
\rightline{({\it A. Golovanov}\/)}
7. In a convex hexagon $AC'BA'CB'$ every two opposite sides are equal.
$A_1$ is the point of intersection of $BC$ with the perpendicular bisector
of $AA'$. $B_1$ and $C_1$ are defined similarly. Prove that
$A_1$, $B_1$, and $C_1$ are collinear.
\rightline{({\it A. Akopyan}\/)}
8. $P(n)$ is a quadratic trinomial with integer coefficients.
For each positive integer $n$ the number $P(n)$ has a proper divisor $d_n$
(i.~e. $11$.
\rightline{({\it A. Golovanov}\/)}
\vfill\eject
\centerline{\it Junior league}
\medskip
\centerline{\tt First day}
\medskip
1. Red, blue, and green children stand in a circle. When a teacher asked
the red children that have a green neighbour to lift a hand, 20 children
lifted their hands. When she asked the blue children that have a green
neighbour to lift a hand, 25 children lifted their hands. Prove that
some child that lifted the hand had two green neighbours.
\rightline{({\it A. Golovanov}\/)}
2. How many ways are there to cut a $11\times11$ square from a
$2011\times2011$ square so that the remained part can be divided into dominoes
($1\times 2$ rectangles)?
\rightline{({\it S. Volchenkov}\/)}
3. The excircle of triangle $ABC$ touches the side $AB$ at $P$ and
the extensions of sides $AC$ and $BC$ at $Q$ and $R$, respectively. Prove
that if the midpoint of $PQ$ lies on the circumcircle of $ABC$ then
the midpoint of $PR$ also lies on that circumcircle.
\rightline{({\it S. Berlov}\/)}
4. Prove that among 100000 consecutive 100-digit numbers there is a number $n$
such that the period length of the decimal expansion of ${1\over n}$
is greater than 2011.
\rightline{({\it A. Golovanov}\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. All the numbers greater than 1 are coloured in two colours (both colours are used). Prove that
there exist real $a$ and $b$ such that the numbers $a+b$ and
$ab$ have different colours.
\rightline{({\it A. Golovanov}\/)}
6. A circle passing through the vertices $A$ and $B$ of a cyclic
quadrilateral $ABCD$ intersects its diagonals $AC$ and $BD$ at $E$ and $F$, respectively.
The lines $AF$ and $BC$ meet at point $P$, the lines $BE$ and $AD$ meet at point $Q$.
Prove that $PQ$ is parallel to $CD$.
\rightline{({\it A. Akopyan}\/)}
7. In a word of more than 10 letters, every two consecutive letters
are different. Prove that one can change places of two consecutive letters
so that the resulting word is not periodic (that is, cannot be divided
into equal subwords).
\rightline{({\it A. Golovanov}\/)}
8. The Duke of Squares left to his three sons a square estate 100 by 100
miles made of ten thousand 1 by 1 mile square plots. To divide the inheritance
he showed each son a point inside the estate and assigned to this son all
the plots such that the distances from their centres to that point are less
than the distances to the points of his brothers. In this way the whole estate
has been divided between the sons. Is it true that, irrespective of the choice of
points, the part assigned to each son is connected (that is, there is a path between every
two of its points, never leaving this part)?
\rightline{({\it A. Akopyan}\/)}
\end