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\centerline{\bf ``TUYMAADA--2009''}
\medskip
\centerline{\it Senior league}
\medskip
\centerline{\tt First day}
\medskip
1. Three real numbers are given. Fractional part of the product of
every two of them is $1\over 2$. Prove that these numbers are irrational.
\rightline{({\it A. Golovanov}\/)}
2. A necklace consists of 100 blue and several red beads. It is known
that every segment of the necklace containing 8 blue beads contain also
at least 5 red beads. What minimum number of red beads can be
in the necklace?
\rightline{({\it A. Golovanov}\/)}
3. On the side $AB$ of a cyclic quadrilateral $ABCD$
there is a point $X$ such that diagonal $BD$ bisects $CX$ and diagonal $AC$
bisects $DX$. What is the minimum possible value of $AB\over CD$?
\rightline{({\it S. Berlov}\/)}
4. Is there a positive integer $n$ such that among 200th digits after
decimal point in the decimal representations of $\sqrt{n}$, $\sqrt{n+1}$,
$\sqrt{n+2}$, \dots, $\sqrt{n+999}$ every digit occurs 100 times?
\rightline{({\it A. Golovanov}\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. A magician asked a spectator to think of a three-digit number
$\overline{abc}$ and then to tell him the sum of numbers $\overline{acb}$, $\overline{bac}$,
$\overline{bca}$, $\overline{cab}$, and $\overline{cba}$. He claims that when he knows
this sum he can determine the original number. Is that so?
\rightline{({\it From olympiad materials}\/)}
6. An arrangement of chips in the squares of $n\times n$ table
is called {\it sparse} if every $2\times 2$ square contains at most 3 chips.
Serge put chips in some squares of the table (one in a square) and
obtained a sparse arrangement.
He noted however that if any chip is moved to any free square then
the arrangement is no more sparce. For what $n$ is this possible?
\rightline{({\it S. Berlov}\/)}
7. A triangle $ABC$ is given. Let $B_1$ be the reflection of $B$ across
the line $AC$,
$C_1$ the reflection of $C$ across the line $AB$, and
$O_1$ the reflection of the circumcentre of $ABC$ across the line $BC$.
Prove that the circumcentre of $ABC$ lies on the line $AO_1$.
\rightline{({\it A. Akopyan}\/)}
8. Determine the maximum number $h$ satisfying the following condition:
for every $a\in [0,h]$ and every polynomial $P(x)$ of degree 99
such that $P(0)=P(1)=0$, there exist $x_1,x_2\in [0,1]$ such that
$P(x_1)=P(x_2)$ and $x_2-x_1=a$.
\rightline{({\it F. Petrov, D. Rostovsky, A. Khrabrov}\/)}
\vfill\eject
\centerline{\it Junior league}
\medskip
\centerline{\tt First day}
\medskip
1. All squares of a $20\times 20$ table are empty. Misha and Sasha in turn
put chips in free squares (Misha begins). The player after whose move
there are four chips on the intersection of two rows and two columns wins.
Which of the players has a winning strategy?
\rightline{({\it A. Golovanov}\/)}
2. $P(x)$ is a quadratic trinomial. What maximum number of terms
equal to the sum of the two preceding terms can occur in the sequence
$P(1)$, $P(2)$, $P(3)$, \dots?
\rightline{({\it A. Golovanov}\/)}
3. In a cyclic quadrilateral $ABCD$ the sides $AB$ and $AD$ are equal,
$CD>AB+BC$. Prove that $\angle ABC>120^\circ$.
%short-list BW-07.
\rightline{({\it From olympiad materials}\/)}
4. Each of the subsets $A_1$, $A_2$, \dots, $A_n$ of a 2009-element set $X$
contains at least 4 elements. The intersection of every two
of these subsets contains at most 2 elements. Prove that
in $X$ there is a 24-element subset $B$ containing neither of the sets
$A_1$, $A_2$, \dots, $A_n$.
\rightline{({\it From olympiad materials}\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. A magician asked a spectator to think of a three-digit number
$\overline{abc}$ and then to tell him the sum of numbers $\overline{acb}$, $\overline{bac}$,
$\overline{bca}$, $\overline{cab}$, and $\overline{cba}$. He claims that when he knows
this sum he can determine the original number. Is that so?
\rightline{({\it From olympiad materials}\/)}
6. $M$ is the midpoint of base $BC$ in a trapezoid $ABCD$. A point $P$
is chosen on the base $AD$. The line $PM$ meets the line $CD$ at a point $Q$
such that $C$ lies between $Q$ and $D$. The perpendicular to the bases
drawn through $P$ meets the line $BQ$ at $K$. Prove that
$\angle QBC = \angle KDA$.
\rightline{({\it S. Berlov}\/)}
7. An arrangement of chips in the squares of $n\times n$ table
is called {\it sparse} if every $2\times 2$ square contains at most 3 chips.
Serge put chips in some squares of the table (one in a square) and
obtained a sparse arrangement.
He noted however that if any chip is moved to any free square then
the arrangement is no more sparce. For what $n$ is this possible?
\rightline{({\it S. Berlov}\/)}
8. The sum of several non-negative numbers is not greater than 200, while
the sum of their squares is not less than 2500. Prove that among them
there are four numbers whose sum is not less than 50.
\rightline{({\it A. Khabrov}\/)}
\end