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\centerline{\bf ``TUYMAADA--2016''}
\medskip
\centerline{\it Senior league}
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\centerline{\tt First day}
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1. The sequence $(a_n)$ is defined by $a_1=0$,
$$
a_{n+1}={a_1+a_2+\ldots+a_n\over n}+1.
$$
Prove that $a_{2016}>{1\over 2}+a_{1000}$.
\rightline{({\it A. Golovanov}\/)}
\smallskip
2. A cube stands on one of the squares of an infinite chessboard.
On each face of the cube there is an arrow pointing in one of the four directions
parallel to the sides of the face. Anton looks on the cube from above and
rolls it over an edge in the direction pointed by the arrow on the top face.
Prove that the cube cannot cover any $5\times 5$ square.
\rightline{({\it A.Chukhnov}\/)}
\smallskip
3. Altitudes $AA_1$, $BB_1$, $CC_1$ of an acute triangle $ABC$
meet at $H$. $A_0$, $B_0$, $C_0$ are the midpoints of $BC$, $CA$, $AB$
respectively. Points $A_2$, $B_2$, $C_2$ on the segments
$AH$, $BH$, $HC_1$ respectively are such that
$\angle A_0B_2A_2 = \angle B_0C_2B_2 = \angle C_0A_2C_2 =90^\circ$.
Prove that the lines $AC_2$, $BA_2$, $CB_2$ are concurrent.
\rightline{({\it A.Pastor}\/)}
\smallskip
4. For each positive integer $k$ determine the number of solutions
of the equation
$$
8^k = x^3+y^3+z^3-3xyz
$$
in non-negative integers $x$, $y$, $z$, $0\!\leq\! x \!\leq\! y\!\leq\! z$.
\rightline{({\it V. Shevelev}\/)}
\medskip
\centerline{\tt Second day}
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5. The ratio of prime numbers $p$ and $q$ does not exceed 2 ($p\ne q$).
Prove that there are two consecutive positive integers such that
the largest prime divisor of one of them is $p$ and that of the other is $q$.
\rightline{({\it A. Golovanov}\/)}
\smallskip
6. The numbers $a$, $b$, $c$, $d$ satisfy $0{3\over 2}$ prove the inequality
$$
x^{24} + \root 5\of {y^{60}+z^{40}} \geq
\left(x^4 y^3 + {1\over 3} y^2 z^2 + {1\over 9} x^3 z^3 \right)^2.
$$
\rightline{({\it K. Kokhas}\/)}
\smallskip
8. A connected graph is given. Prove that its vertices can be coloured
blue and green and some of its edges marked so that every two vertices
are connected by a path of marked edges, every marked edge connects
two vertices of different colour and no two green vertices are connected
by an edge of the original graph.
\rightline{({\it V. Dolnikov}\/)}
\vfill\eject
\centerline{\it Junior league}
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\centerline{\tt First day}
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1. Tanya and Serezha have a heap of 2016 candies. They make moves in turn,
Tanya moves first. At each move a player can eat either one candy or (if
the number of candies is even at the moment) exactly half of all candies.
The player that cannot move loses. Which of the players has a winning strategy?
\rightline{({\it A. Golovanov}\/)}
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2. The point $D$ on the altitude $AA_1$ of an acute triangle $ABC$ is such that
$\angle BDC=90^\circ$; $H$ is the orthocentre of $ABC$. A circle
with diameter $AH$ is constructed. Prove that the tangent drawn from $B$
to this circle is equal to $BD$.
\rightline{({\it L.Emelyanov}\/)}
\smallskip
3. A cube stands on one of the squares of an infinite chessboard.
On each face of the cube there is an arrow pointing in one of the four directions
parallel to the sides of the face. Anton looks on the cube from above and
rolls it over an edge in the direction pointed by the arrow on the top face.
Prove that the cube cannot cover any $5\times 5$ square.
\rightline{({\it A.Chukhnov}\/)}
\smallskip
4. Non-negative numbers $a$, $b$, $c$ satisfy
$a^2+b^2+c^2\geq 3$. Prove the inequality
$$
(a+b+c)^3\geq 9(ab+bc+ca).
$$
\rightline{({\it A.Khrabrov}\/)}
\medskip
\centerline{\tt Second day}
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5. Positive numbers are written in the squares of a $10\times 10$ table.
Frogs sit in five squares and cover the numbers in these squares. Kostya
found the sum of all visible numbers and got 10. Then each frog jumped to an
adjacent square and Kostya's sum changed to $10^{2}$. Then the frogs
jumped again, and the sum changed to $10^3$ and so on: every new sum was
10 times greater than the previous one. What maximum sum can Kostya
obtain?
\rightline{({\it K. Kokhas}\/)}
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6. Is there a positive integer $N>10^{20}$ such that all its decimal digits are odd,
the numbers of digits 1, 3, 5, 7, 9 in its decimal representation are equal,
and it is divisible by each 20-digit number obtained from it by deleting
digits? (Neither deleted nor remaining digits must be consecutive.)
\rightline{({\it S. Berlov}\/)}
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7. The numbers $a$, $b$, $c$, $d$ satisfy $0