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\input amssym.def
\input amssym.tex
\centerline{\bf ``TUYMAADA--2021''}
\medskip
\centerline{\it Senior league}
\medskip
\centerline{\tt First day}
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1. Polynomials $F$ and $G$ satisfy
$$
F(F(x)) > F(G(x)) > G(G(x))
$$
for all real $x$. Prove that $F(x) > G(x)$ for all real $x$.
\medskip
\rightline{({\it V.~Frank}\/)}
\smallskip
2. In trapezoid $ABCD$, $M$ is the midpoint of base $AD$. Point $E$
lies on the segment $BM$. It is known that
$\angle ADB = \angle MAE = \angle BMC$. Prove that the triangle $BCE$ is isosceles.
\rightline{({\it A.~Kuznetsov}\/)}
\smallskip
3. Positive real numbers $a_1$, \dots, $a_k$, $b_1$, \dots, $b_k$ are given.
Let $A=\sum\limits_{i=1}^k a_i$, $B=\sum\limits_{i=1}^k b_i$.
Prove the inequality
$$\left(\sum_{i=1}^k {a_ib_i\over a_iB+b_iA} -1\right)^2\geq
\sum_{i=1}^k {a_i^2\over a_iB+b_iA} \cdot \sum_{i=1}^k {b_i^2\over a_iB+b_iA}.$$
\rightline{({\it F.~Dong, J.~Ge }\/)}
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4. An $n\times n$ square ($n$ is a positive integer) consists of $n^2$
unit squares. {\it A monotonous path} in this square is a path of length $2n$
beginning in the left lower corner of the square, ending in its right upper
corner and going along the sides of unit squares.
For each $k$, ${0\le k < 2n-1}$, let $S_k$ be the set of all the monoto\-nous
paths such that the number of unit squares lying below the path leaves
remainder $k$ upon division by $2n-1$. Prove that all $S_k$ contain
equal number of elements.
\rightline{({\it M.~Just, M. Schneider}\/)}
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\centerline{\tt Second day}
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5. Sines of three acute angles form an arithmetic progression,
while the cosines of these angles form a geometric progression.
Prove that all three angles are equal.
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\rightline{({\it S.~Berlov}\/)}
6. In a $n\times n$ table ($n>1$) $k$ unit squares are marked.
One wants to rearrange rows and columns so that all the marked unit squares
are above the main diagonal or on it. For what maximum $k$ is it always possible?
\rightline{({\it M.~Antipov}\/)}
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7. An acute triangle $ABC$ is given, $AC \ne BC$. The altitudes drawn
from $A$ and $B$ meet at $H$ and intersect the external bisector of the angle $C$
at $Y$ and $X$ respectively. The external bisector of the angle $AHB$
meets the segments $AX$ and $BY$ at $P$ and $Q$ respectively. If
$PX=QY$, prove that $AP+BQ\geq 2CH$.
\rightline{({\it E.~Lopatin, D.~Shiryaev}\/)}
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8. In a sequence $P_n$ of quadratic trinomials each trinomial, starting
with the third, is the sum of the two preceding trinomials. The first two
trinomials do not have common roots. Is it possible that $P_n$ has an integral
root for each $n$?
\rightline{({\it A.~Golovanov}\/)}
\vfill\eject
\centerline{\it Junior league}
\medskip
\centerline{\tt First day}
\medskip
1. Quadratic trinomials $F$ and $G$ satisfy
$$
F(F(x)) > F(G(x)) > G(G(x))
$$
for all real $x$. Prove that $F(x) > G(x)$ for all real $x$.
\rightline{({\it V.~Frank}\/)}
\smallskip
2. The bisector of angle $B$ of a parallelogram $ABCD$ meets its diagonal
$AC$ at $E$, and the external bisector of angle $B$ meets line $AD$ at $F$.
$M$ is the midpoint of $BE$. Prove that $CM \parallel EF$.
\rightline{({\it A.~Kuznetsov}\/)}
\smallskip
3. For $n$ distinct positive integers all their $n(n-1)/2$ pairwise sums
are considered. For each of these sums Ivan has written on the board
the number of original integers which are less than that sum and divide it.
What is the maximum possible sum of the numbers written by Ivan?
\rightline{({\it S.~Berlov}\/)}
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4. Some manors of Lipshire county are connected by roads. The inhabitants
of manors connected by a road are called {\it neighbours}. Is it always
possible to settle in each manor a knight (who always tells truth) or a liar
(who always lies) so that every inhabitant can say "The number of liars
among my neighbours is at least twice the number of knights"?
\rightline{({\it S.~Berlov}\/)}
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\centerline{\tt Second day}
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5. In a $100\times 100$ table 110 unit squares are marked. Is it
always possible to rearrange rows and columns so that all the marked unit
squares are above the main diagonal or on it?
\rightline{({\it M.~Antipov}\/)}
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6. Given are real $y>1$ and positive integer $n\leq y^{50}$
such that all prime divisors of $n$ do not exceed $y$.
Prove that $n$ is a product of $99$ positive integer factors
(not necessarily primes) not exceeding $y$.
\rightline{({\it G.~Martin, A.~Parvardi }\/)}
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7. A pile contains $2021^{2021}$ stones. In a move any pile can be divided
into two piles so that the numbers of stones in them differ by a power of 2
with non-negative integer exponent. After some move it turned out that
the number of stones in each pile is a power of 2 with non-negative integer
exponent. Prove that the number of moves performed was even.
\rightline{({\it M.~Antipov}\/)}
\smallskip
8. See problem 7 of Senior league.
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