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\centerline{\bf ``TUYMAADA--2008''}
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\centerline{\it Senior league}
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\centerline{\tt First day}
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1. Several irrational numbers are written on a blackboard. It is known
that for every two numbers $a$ and $b$ on the blackboard, at
least one of the numbers $a\over b+1$ and $b\over a+1$ is rational.
What maximum number of irrational numbers can be on the blackboard?
\rightline{({\it A. Golovanov}\/)}
2. Is it possible to arrange on a circle all composite positive integers
not exceeding $10^6$, so that no two neighbouring numbers are coprime?
\rightline{({\it A. Golovanov}\/)}
3. Point $I_1$ is the reflection of incentre $I$ of triangle $ABC$ across
the side $BC$. The circumcircle of $BCI_1$ intersects the line $II_1$
again at point $P$. It is known that $P$ lies outside the incircle
of the triangle $ABC$. Two tangents drawn from $P$ to the latter circle
touch it at points $X$ and $Y$. Prove that the line $XY$ contains a
medial line of the triangle $ABC$.
\rightline{({\it L. Emelyanov}\/)}
4. A group of persons is called {\it good} if its members can be distributed
to several rooms so that nobody is acquainted with any person in the same room
but it is possible to choose a person from each room so that all
the chosen persons are acquainted with each other.
A group is called {\it perfect} if it is good and every set of its members
is also good.
A perfect group planned a party. However one of its members, Vasya,
brought his acquaintance Petya, who was not originally expected,
and introduced him to all his other acqaintances. Prove that
the new group is also perfect.
\rightline{({\it C. Berge}\/)}
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\centerline{\tt Second day}
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5. Every street in the city of Hamiltonville connects two squares, and
every square may be reached by streets from every other.
The governor discovered that if he closed all squares of any route not
passing any square more than once, every remained square would be
reachable from each other.
Prove that there exists a circular route passing every square of the
city exactly once.
\rightline{({\it S. Berlov}\/)}
6. A set $X$ of positive integers is called {\it nice} if for each pair
$a$, $b\in X$ exactly one of the numbers $a+b$ and $|a-b|$ belongs to
$X$ (the numbers $a$ and $b$ may be equal). Determine the number
of nice sets containing the number 2008.
\rightline{({\it F. Petrov}\/)}
7. A loader has two carts. One of them can carry up to 8 kg, and another
can carry up to 9 kg. A finite number of sacks with sand lie in a
storehouse. It is known that their total weight is more than 17 kg, while
each sack weighs not more than 1 kg. What maximum weight of sand can the
loader carry on his two carts, regardless of particular weights of sacks?
\rightline{({\it M.Ivanov, D.Rostovsky, V.Frank}\/)}
8. A convex hexagon is given. Let $s$ be the sum of the lengths of
the three segments connecting the midpoints of its opposite sides.
Prove that there is a point in the hexagon such that the sum of its distances
to the lines containing the sides of the hexagon does not exceed $s$.
\rightline{({\it N. Sedrakyan}\/)}
\vfill\eject
\centerline{\it Junior league}
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\centerline{\tt First day}
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1.
Portraits of famous scientists hang on a wall. The scientists lived
between 1600 and 2008, and none of them lived longer than 80 years. Vasya
multiplied the years of birth of these scientists, and Petya multiplied
the years of their death. Petya's result is exactly $5\over 4$ times
greater than Vasya's result. What minimum number of portraits can be
on the wall?
\rightline{({\it V. Frank}\/)}
2.
Prove that all composite positive integers not exceeding $10^6$
may be arranged on a circle so that no two neighbouring numbers are coprime.
\rightline{({\it A. Golovanov}\/)}
3. 100 unit squares of an infinite squared plane form a $10\times 10$ square.
Unit segments forming these squares are coloured in several colours.
It is known that the border of every square with sides on grid lines
contains segments of at most
two colours.
(Such square is not necessarily contained in the original $10\times 10$
square.)
What maximum number of colours may appear in this colouring?
\rightline{({\it S. Berlov}\/)}
4.
Point $I_1$ is the reflection of incentre $I$ of triangle $ABC$ across
the side $BC$. The circumcircle of $BCI_1$ intersects the line $II_1$
again at point $P$. It is known that $P$ lies outside the incircle
of the triangle $ABC$. Two tangents drawn from $P$ to the latter circle
touch it at points $X$ and $Y$. Prove that the line $XY$ contains the
medial line of the triangle $ABC$.
\rightline{({\it L. Emelyanov}\/)}
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\centerline{\tt Second day}
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5.
A loader has a waggon and a little cart. The waggon can carry up to
1000~kg, and the cart can carry only up to 1~kg.
A finite number of sacks with sand lie in a storehouse.
It is known that their total weight is more than 1001 kg, while each
sack weighs not more than 1 kg. What maximum weight of
sand can the loader carry in the waggon and the cart, regardless of
particular weights of sacks?
\rightline{({\it M.Ivanov, D.Rostovsky, V.Frank}\/)}
6. Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$.
Its diagonals $AC$ and $BD$ intersect at point $M$. Points $X$ and $Y$
on the segment $AB$ are such that $AX=AM$, $BY=BM$. Let $Z$ be the
midpoint of $XY$ and $N$ is the point of intersection of the segments
$XD$ and $YC$. Prove that the line $ZN$ is parallel to the bases
of the trapezoid.
\rightline{({\it A. Akopyan, A. Myakishev}\/)}
7.
A set $X$ of positive integers is called {\it nice} if for each
$a$, $b\in X$ exactly one of the numbers $a+b$ and $|a-b|$ belongs to
$X$ (the numbers $a$ and $b$ may be equal). Determine the number
of all nice sets containing the number 2008.
\rightline{({\it F. Petrov}\/)}
8. 250 numbers are chosen among positive integers not exceeding 501.
Prove that for every integer $t$ there are four chosen
numbers $a_1$, $a_2$, $a_3$, $a_4$, such that
$a_1 + a_2 + a_3 + a_4 -t$ is divisible by 23.
\rightline{({\it K. Kokhas}\/)}
\end