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\centerline{\bf ``TUYMAADA--2016''}
\medskip
\centerline{\it Senior league}
\medskip
\centerline{\tt First day}
\medskip
1. Functions $f$ and $g$ are defined on the set of all integers
in the interval $[-100; 100]$ and take integral values.
Prove that for some integral $k$ the number of solutions of the equation
$$f(x)-g(y)=k$$ is odd.
%(А.Голованов)
\rightline{({\it A.~Golovanov}\/)}
\smallskip
2. The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$
are perpendicular and meet at point $P$. The point $Q$ on the segment $PC$
is such that $AP=QC$. Prove that the perimeter of the triangle $BQD$
is at least $2AC$.
\rightline{({\it A.~Kuznetsov}\/)}
\smallskip
3. All the sides of a right triangle with area $S$ are rational.
Prove that there exists a right triangle
not equal to the original one
such that all its sides are rational and its area is $S$.
\rightline{({\it S.~Chan}\/)}
\smallskip
4. There are 25 masks of different colours. $k$ sages play the
following game. They are shown all the masks. Then the sages agree
on their strategy. After that the masks are put on them so that
each sage sees the masks on the others but can not see who wears each mask
and does not see his own mask. No communication is allowed.
Then each of them simultaneously names one colour trying to guess the colour of his mask.
Find the minimum $k$ for which the sages
can agree so that at least one of them surely guesses the colour of his mask.
\rightline{({\it S.~Berlov }\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. Does there exist a quadratic trinomial $f(x)$ such that
$f(1/2017)=1/2018$, $f(1/2018)=1/2017$, and two of its coefficients
are integers?
\rightline{({\it A. Khrabrov}\/)}
\smallskip
6. Let $\sigma(n)$ denote the sum of positive divisors of a number~$n$.
A positive integer $N=2^r b$ is given, where $r$ and $b$ are positive integers
and $b$ is odd. It is known that $\sigma(N)=2N-1$. Prove that $b$ and
$\sigma(b)$ are coprime.
\rightline{({\it J. Antalan, J. Dris}\/)}
\smallskip
7. A point $E$ lies on the extension of the side $AD$ of the rectangle
$ABCD$ over $D$. The ray $EC$ meets the circumcircle $\omega$ of $ABE$
at the point $F\ne E$. The rays $DC$ and $AF$ meet at $P$. $H$ is the
foot of the perpendicular drawn from $C$ to the line $\ell$ going through
$E$ and parallel to $AF$. Prove that the line $PH$ is tangent to
$\omega$.
\rightline{({\it A. Kuznetsov}\/)}
\smallskip
8. Two points $A$ and $B$ are given in the plane. A point ~$X$ is called
their {\it preposterous midpoint} if there is a Cartesian coordinate
system in the plane such that the coordinates of $A$ and $B$ in this system
are non-negative, the abscissa of $X$ is the geometric mean of the abscissae
of $A$ and $B$, and the ordinate of $X$ is the geometric mean of the ordinates
of $A$ and $B$. Find the locus of all the preposterous midpoints of $A$ and $B$.
\rightline{({\it K. Tyschuk}\/)}
\vfill\eject
\centerline{\it Junior league}
\medskip
\centerline{\tt First day}
\medskip
1. Functions $f$ and $g$ are defined on the set of all integers
in the interval $[-100; 100]$ and take integral values.
Prove that for some integral $k$ the number of solutions of the equation
$$f(x)-g(y)=k$$ is odd.
%(А.Голованов)
\rightline{({\it A.~Golovanov}\/)}
\smallskip
2. The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$
are perpendicular and meet at point $P$. The point $Q$ on the segment $PC$
is such that $AP=QC$. Prove that the perimeter of the triangle $BQD$
is at least $2AC$.
\rightline{({\it A.~Kuznetsov}\/)}
\smallskip
3. Every two cities in a country are connected either by a direct bus route
or by a direct plane flight. {\it A clique} is a set of cities such that
every two of them are connected by a direct flight. {\it A cluque}
is a set of cities such that every two of them are connected by a direct flight,
and the numbers of bus routes starting in each of them are equal.
{\it A claque} is a set of cities such that every two of them are connected
by a direct flight, and the numbers of bus routes starting in all of them are
different. Prove that the the number of cities in every clique does not
exceed the product of the largest possible number of cities in a cluque and
the largest possible number of cities in a claque.
\rightline{({\it P.~Borg, Y.~Caro, translated by K.~Kokhas}\/)}
\smallskip
4. All the sides of a right triangle with area $S$ are rational.
Prove that there exists a right triangle
not equal to the original one
such that all its sides are rational and its area is $S$.
\rightline{({\it S.~Chan}\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. $BL$ is the bisector of an isosceles triangle $ABC$. A point $D$ is
chosen on the base $BC$ and a point $E$ is chosen on the lateral side
$AB$ so that $AE ={1\over 2}AL=CD$. Prove that $LE=LD$.
\rightline{({\it A. Kuznetsov}\/)}
\smallskip
6. Let $\sigma(n)$ denote the sum of positive divisors of a number~$n$.
A positive integer $N=2^r b$ is given, where $r$ and $b$ are positive integers
and $b$ is odd. It is known that $\sigma(N)=2N-1$. Prove that $b$ and
$\sigma(b)$ are coprime.
\rightline{({\it J. Antalan, J. Dris}\/)}
\smallskip
7. An equilateral triangle with side 20 is divided by three series
of parallel lines into 400 equilateral triangles with side 1.
What maximum number of these small triangles can be crossed (internally)
by one line?
\rightline{({\it A. Golovanov}\/)}
\smallskip
8. Consider a graph with vertices $A_1$, $A_2$, \dots, $A_{2017}$,
$B_1$, $B_2$, \dots, $B_{2017}$ and edges
$A_iB_i$, $A_iA_{i+1}$, $B_iB_{i+17}$ (in cyclic numbering).
Is it true that 4 cops can catch one robber on this graph for every initial
position of cops and robber?
(First all the cops make their moves, then robber makes his move, then again
all the cops make their moves, etc.
In a move, a person can stay in his/her vertex or jump to any of the
neighboring vertices.
Everybody knows about positions of all others. The cops can coordinate their moves.
The robber is caught if after some move he shares his vertex with some cop).
\rightline{({\it T.~Ball, R.~Bell, J.~Guzman, M.~Hanson-Colvin, N.~Schonsheck}\/)}
\end