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\centerline{\bf ``TUYMAADA--2010''}
\medskip
\centerline{\it Senior league}
\medskip
\centerline{\tt First day}
\medskip
1. Sasha and Dima play on a $100\times 100$ board. In the beginning
Sasha chooses 50 squares and puts one king on each of them. After that
Dima chooses one free square and places a rook on it. Then the players
make their moves in turn (Sasha begins). At each move Sasha moves each king
to an adjacent square (horizontally, vertically or diagonally) and Misha
moves the rook horizontally or vertically by any number of squares. The
rook cannot jump over a king or capture it. Can Sasha play so that one of
the kings eventually captures the rook?
\rightline{({\it S. Berlov}\/)}
2. $H$ is the orthocentre of an acute triangle $ABC$. A point $D$
is chosen within the side $BC$. The point $P$ is constructed so that
$ADPH$ is a parallelogram. Prove that $\angle DCP<\angle BHP$.
\rightline{({\it S. Berlov}\/)}
3. On a circle 2010 digits are arranged, each equal to 1, 2, or 3.
It is known that for every $k$ each of the digits appears at most $k+10$ times
in any block of $3k$ consecutive digits. Prove that there is a block of
several consecutive digits where every digits appears the same number of times.
\rightline{({\it S. Berlov}\/)}
4. Prove that for every real $\alpha>0$ the number $[\alpha n^2]$
is even for infinitely many positive integers $n$.
\rightline{({\it A. Golovanov}\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. Baron Munchhausen boasts that he knows a remarkable quadratic
triniomial with positive coefficients. The trinomial itself has an
integral root; if all its coefficients are increased by 1, the
trinomial obtained also has an integral root; and if
all its coefficients are again increased by 1, the new trinomial, too,
has an integral root. Can this be true?
\rightline{({\it S. Berlov}\/)}
6. A positive integer $n$ is given. It is known that there exist 2010
consecutive positive integers such that none of them is divisible by $n$
but their product is divisible by $n$.
Prove that there exist 2004 consecutive positive integers such that none
of them is divisible by $n$ but their product is divisible by $n$.
\rightline{({\it S. Berlov}\/)}
7. The extensions of sides $AB$ and $CD$ of a cyclic quadrilateral
$ABCD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet
at point $Q$. Prove that the distance between the orthocentres of
triangles $APD$ and $AQB$ is equal to the distance between the orthocentres
of triangles $CQD$ and $BPC$.
\rightline{({\it L. Emelyanov}\/)}
8. In a country there are $4^9$ schoolchildren living in four cities.
At the end of the school year a state examination was held in 9 subjects.
It is known that any two students have different marks at least in one
subject. However, every two students from the same city got equal marks
at least in one subject. Prove that there is a subject such that every two
children living in the same city have equal marks in this subject.
\rightline{({\it F. Petrov}\/)}
\vfill\eject
\centerline{\it Junior league}
\medskip
\centerline{\tt First day}
\medskip
1. Sasha and Dima play on a $100\times 100$ board. In the beginning
Sasha chooses 50 squares and puts one king on each of them. After that
Dima chooses one free square and places a rook on it. Then the players
make their moves in turn (Sasha begins). At each move Sasha moves each king
to an adjacent square (horizontally, vertically or diagonally) and Misha
moves the rook horizontally or vertically by any number of squares. The
rook cannot jump over a king or capture it. Can Sasha play so that one of
the kings eventually captures the rook?
\rightline{({\it S. Berlov}\/)}
2. $H$ is the orthocentre of an acute triangle $ABC$. A point $D$
is chosen within the side $BC$. The point $P$ is constructed so that
$ADPH$ is a parallelogram. Prove that $\angle BPC>\angle BAC$.
\rightline{({\it S. Berlov}\/)}
3. Three different non-zero numbers are given. Every quadratic
trinomial whose coefficients are these three numbers in any order
has an integral root. Prove that all such trinomials have root 1.
\rightline{({\it A. Golovanov}\/)}
4. On a blackboard 2010 positive integers are written. One can remove
any two numbers $x$, $y$ with $y>1$ and write either $2x+1$, $y-1$ or
$2x+1$, ${1\over 4}(y-1)$ (if 4 divides $y-1$) instead. For instance, one can
remove 3 and 5 and write 7 and 4, or 7 and 1 (taking $x=3$, $y=5$), or
11, 2 (taking $x=5$, $y=3$). This was done several times; In the beginning
2006 and 2008 were removed. Prove that the original set of numbers can not
appear again in the blackboard.
\rightline{({\it M. Antipov}\/)}
\medskip
\centerline{\tt Second day}
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5. A set $M$ of real numbers contains more than one element. It is known
that for every $x$ in $M$ at least one of the numbers $3x-2$ and $-4x+5$
also belongs to $M$. Prove that $M$ is infinite.
\rightline{({\it A. Golovanov}\/)}
6. A positive integer $n$ is given. It is known that there exist five
consecutive positive integers such that none of them is divisible by $n$
but their product is divisible by $n$. Prove that there exist four
consecutive positive integers such that none of them is divisible by $n$
but their product is divisible by $n$.
\rightline{({\it S. Berlov}\/)}
7. A triangle $ABC$ is given. Perpendicular $IP$ is drawn from its
incentre $I$ to the line passing through $A$ parallel to $BC$.
The tangent to the incircle parallel to $BC$ meets the sides $AB$ and $AC$
at $Q$ and $R$ respectively.
Prove that $\angle QPB=\angle RPC$.
\rightline{({\it V. Smykalov}\/)}
8. There are several cities in a country. Direct one-way flights run
between some cities. Prove that one can select a group $A$ of cities
such that
(i) there is no flight between any two cities of $A$;
(ii) from every city outside $A$ one can reach some city of $A$ either
by direct flight or by two flights with one change.
\rightline{({\it V. Dolnikov\/})}
\end