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\input amssym.def
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\centerline{\bf ``TUYMAADA--2015''}
\medskip
\centerline{\it Senior league}
\medskip
\centerline{\tt First day}
\medskip
1. On a playing field $n$ footballers practiced, all of them were forwards
or goalkeepers. $k$ goals were scored during the exercise in all. Prove that
Don Fabio can assign numbers from 1 to $n$ to all footballers so that
for each goal the numbers of the forward and the goalkeeper differ at least by
$n-k$.
\rightline{({\it S.~Berlov}\/)}
\smallskip
2. Median $BD$ is drawn in a triangle $ABC$. Bisectors of the angles
$ABD$ and $ACB$ are perpendicular. Determine the maximum possible value
of the angle $BAC$.
\rightline{({\it S.~Berlov}\/)}
\smallskip
3. Polynomial $P(x,y)$ with real coefficients satisfies
$$P(x+2y, x+y)=P(x,y).$$
Prove that $P(x,y)=Q\bigl((x^2-2y^2)^2\bigr)$
for some polynomial $Q(t)$.
\rightline{({\it A.~Golovanov}\/)}
\smallskip
4. The number $n!$ is written in the form $ab^2$, where $a$ is squarefree.
Prove that for every $\varepsilon>0$ the inequality
$$2^{(1-\varepsilon)n}14$.
Prove that not all positive integers $n$ can be written in the form
$$
n=x(ax + b) + y(ay + c) + z(az + d),
$$
with integral $x$, $y$, $z$.
\rightline{({\it K. Kokhas}\/)}
\smallskip
7. In a triangle $ABC$ \ $M$ is the midpoint of $AB$ and $O$ is the
circumcentre. It is known that $R-r=OM$. The external bisector of angle $A$
meets the line $BC$ at $D$, and the external bisector of angle $C$
meets the line $AB$ at $E$. Determine all possible values of the angle $CED$.
\rightline{({\it D. Shiryaev}\/)}
\smallskip
8. In the plane $k(k+1)/2+1$ points are marked. Some of these points
are joined by non-intersecting segments that can not pass throw the marked point so that the plane is divided
into several parallelograms and one infinite region. What maximum number
of segments can be drawn?
\rightline{({\it A. Kupavsky, A.~Polyansky}\/)}
\vfill\eject
\centerline{\it Junior league}
\medskip
\centerline{\tt First day}
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1. 100 different real numbers are given. Prove that they can be arranged
in the squares of $10\times 10$ table so that no two numbers in squares
sharing a side differ by 1.
\rightline{({\it A.~Golovanov}\/)}
\smallskip
2. A positive integer is called {\it funny} if for some $k$
the sum of its digits equals $k$ and number itself is divisible by $k+1$.
What maximum number of consecutive integers can be funny?
\rightline{({\it O.~Podlipsky}\/)}
\smallskip
3. Median $BD$ is drawn in a triangle $ABC$. Bisectors of the angles
$ABD$ and $ACB$ are perpendicular. Determine the maximum possible value
of the angle $BAC$.
\rightline{({\it S.~Berlov}\/)}
\smallskip
4. Prove that there exists a positive integer $n$ such that in the
decimal representation of each of the numbers $\sqrt{n}$,
$\root 3\of n$, $\root 4\of n$, \dots,
$\root {10}\of n$ digits 2015\dots\ stand immediately after
the decimal point.
\rightline{({\it A.~Golovanov}\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. A positive integer is increased by its largest proper divisor, the number
obtained is increased by its largest proper divisor and so on.
Prove that after several operations a number divisible by $3^{2000}$ is
obtained.
\rightline{({\it A. Golovanov}\/)}
\smallskip
6. Does there exist an increasing sequence $(a_n)$ of positive
integers such that each positive integer occurs exactly once among
the numbers $a_{n+1}-a_n$, and each positive integer not less than
some number occurs exactly once among
the numbers $a_{n+2}-a_n$?
\rightline{({\it A. Golovanov}\/)}
\smallskip
7. The extenson of bisector $CL$ of a triangle $ABC$ meets the circumcircle
of the triangle at point $K$. $I$ is the incentre of the triangle.
It is known that $IL=LK$. Prove that $CI=IK$.
\rightline{({\it D. Shiryaev}\/)}
\smallskip
8. Four sages stand around a non-transparent baobab. Each of the sages
wears red, blue, or green hat. A sage sees only his two neighbours.
Each of them at the same time must make a guess about the colour of his hat.
If at least one sage guesses correctly, the sages win. They could consult
before the game started. How should they act to win?
\rightline{({\it K. Kokhas}\/)}
\end