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\centerline{\bf ``TUYMAADA--2018''}
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\centerline{\it Senior league}
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\centerline{\tt First day}
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1. Do there exist three different quadratic trinomials $f(x)$,
$g(x)$, $h(x)$ such that the roots of the equation $f(x)=g(x)$ are 1
and 4, the roots of the equation $g(x)=h(x)$ are 2 and 5, and the
roots of the equation $h(x)=f(x)$ are 3 and 6?
\rightline{({\it A.~Golovanov}\/)}
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2. 2550 rooks and $k$ pawns are arranged on a $100\times 100$
board. The rooks cannot leap over pawns. For which minimum $k$ it is
possible that no rook can capture any other rook?
\rightline{({\it N.~Vlasova}\/)}
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3. A point $P$ on the side $AB$ of a triangle $ABC$ and points $S$
and $T$ on the sides $AC$ and $BC$ are such that $AP=AS$ and
$BP=BT$. The circumcircle of $PST$ meets the sides $AB$ and $BC$
again at $Q$ and $R$, respectively. The lines $PS$ and $QR$ meet at
$L$. Prove that the line $CL$ bisects the segment $PQ$.
\rightline{({\it A.~Antropov}\/)}
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4. Prove that for every positive integers $d > 1$ and $m$ the
sequence $a_n = 2^{2^n}+ d$ contains two terms $a_k$ and~$a_\ell$
($k\ne \ell$) such that their greatest common divisor is greater
than $m$.
\rightline{({\it T.~Hakobyan }\/)}
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\centerline{\tt Second day}
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5. A prime $p$ and a positive integer $n$ are given.
The product
$$
(1^3+1)(2^3+1)\ldots ((n-1)^3+1)(n^3+1)
$$
is divisible by $p^3$. Prove that $p\leq n+1$.
\rightline{({\it Z. Luria}\/)}
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6. Prove the inequality
$$
(x^3+2y^2+3z)(4y^3+5z^2+6x)(7z^3+8x^2+9y) \geq 720(xy+yz+xz)
$$
for $x$, $y$, $z\geq 1$.
\rightline{({\it K. Kokhas}\/)}
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7. A school has three senior classes of $M$ students each.
Every student knows at least ${3\over 4} M$ people in each of the two other classes.
Prove that the school can send $M$ non-intersecting teams to the olympiad so that each
team consists of 3 students from different classes who know each other.
\rightline{({\it C.~Magyar, R.~Martin}\/)}
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8. Quadrilateral $ABCD$ with perpendicular diagonals is inscribed in
a circle with centre $O$. The tangents to this circle at $A$ and $C$ together
with line $BD$ form the triangle $\Delta$.
Prove that the circumcircles of $BOD$ and $\Delta$ are tangent.
\rightline{({\it A. Kuznetsov}\/)}
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\centerline{\it Junior league}
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\centerline{\tt First day}
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1. Real numbers $a\ne 0$, $b$, $c$ are given. Prove that there is
a polynomial $P(x)$ with real coefficients such that the polynomial
$x^2+1$ divides the polynomial $aP^2(x)+bP(x)+c$.
\rightline{({\it A.~Golovanov}\/)}
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2. A circle touches the side $AB$ of the triangle $ABC$ at $A$,
touches the side $BC$ at $P$ and intersects the side $AC$ at $Q$.
The line symmetrical to $PQ$ with respect to $AC$ meets the line
$AP$ at $X$. Prove that $PC=CX$.
\rightline{({\it S.~Berlov}\/)}
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3. 2551 rooks and $k$ pawns are arranged on a $100\times 100$
board. The rooks cannot leap over pawns. For which minimum $k$ it is
possible that no rook can capture any other rook?
\rightline{({\it A.~Kuznetsov}\/)}
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4. Prove that for every odd positive integer $d > 1$ and every
positive integer $m$ the sequence $a_n = 2^{2^n}+ d$ contains two
terms $a_k$ and~$a_\ell$ ($k\ne \ell$) such that their greatest
common divisor is greater than $m$.
\rightline{({\it T.~Hakobyan }\/)}
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\centerline{\tt Second day}
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5. 99 identical balls lie on a table. 50 balls are made of copper, and 49 balls
are made of zinc. The assistant numbered the balls.
One spectrometer test is applied to 2 balls and allows to determine whether
they are made of the same metal or not. However, the results of the test
can be obtained only the next day.
What minimum number of tests is required to determine the material of each
ball if all the tests should be performed today?
\rightline{({\it N. Vlasova, S. Berlov}\/)}
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6. The numbers 1, 2, 3, \dots, 1024 are written on a blackboard. They
are divided into pairs. Then each pair is wiped off the board and
non-negative difference of its numbers is written on the board instead.
512 numbers obtained in this way are divided into pairs and so on.
One number remains on the blackboard after ten such operations.
Determine all its possible values.
\rightline{({\it A. Golovanov}\/)}
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7. Prove the inequality
$$
(x^3+2y^2+3z)(4y^3+5z^2+6x)(7z^3+8x^2+9y) \geq 720(xy+yz+xz)
$$
for $x$, $y$, $z\geq 1$.
\rightline{({\it K. Kokhas}\/)}
\smallskip
8. Quadrilateral $ABCD$ with perpendicular diagonals is inscribed in
a circle with centre $O$. The tangents to this circle at $A$ and $C$ together
with line $BD$ form the triangle $\Delta$. Let $\omega$ be the circumcircle
of $OAC$. Prove that the circumcircles of $BOD$ and $\Delta$ are tangent
and their common point belongs to $\omega$.
\rightline{({\it A. Kuznetsov}\/)}
\end