\nopagenumbers
\input amssym.def
\input amssym.tex
\def\R{\Bbb R}
\centerline{\bf ``TUYMAADA--2022''}
\medskip
\centerline{\it Senior league}
\medskip
\centerline{\tt First day}
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1. Some of 100 towns of a kingdom are connected by roads. It is known
that for each two towns $A$ and $B$ connected by a~road there is a town $C$
which is not connected by a road with at least one of the towns $A$ and $B$.
Determine the maximum possible number of roads in the kingdom.
\rightline{({\it P.~Qiaoa, X.~Zhan }\/)}
\smallskip
2. Two circles $\omega_1$ and $\omega_2$ of different radii touch externally
at~$L$. A~line touches $\omega_1$ at $A$ and $\omega_2$ at $B$
(the points $A$ and $B$ are different from $L$). A point $X$ is chosen
in the plane. $Y$ and $Z$ are the second points of intersection
of the lines $XA$ and $XB$ with $\omega_1$ and $\omega_2$ respectively.
Prove that all $X$ such that $AB \parallel YZ$ belong to one circle.
\rightline{({\it K.~Ivanov}\/)}
\smallskip
3. Is there a colouring of all positive integers in three colours so that
for each positive integer the numbers of its divisors of any two colours
differ at most by 2?
\rightline{({\it A.~Golovanov}\/)}
\smallskip
4. For every positive $a_1$, $a_2$, $\dots$, $a_6$ prove the inequality
$$
\root 4\of {a_1\over a_2+a_3+a_4}+
\root 4\of {a_2\over a_3+a_4+a_5}+\ldots+
\root 4\of {a_6\over a_1+a_2+a_3} \ge 2.
$$
\rightline{({\it A. Khrabrov}\/)}
\medskip
\centerline{\tt Second day}
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5. Prove that a quadratic trinomial $x^2+ax+b$ ($a, b\in \R$)
cannot attain at ten consecutive integral points values equal to powers of 2
with non-negative integral exponent.
\rightline{({\it F.~Petrov}\/)}
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6. Kostya marked the points $A(0,1)$, $B(1,0)$, $C(0,0)$ in the
coordinate plane. On the legs of the triangle $ABC$ he marked the points
with coordinates $({1\over 2}, 0)$, $({1\over 3}, 0)$, \dots, $({1\over n+1}, 0)$
and $(0, {1\over 2})$, $(0, {1\over 3})$, \dots, $(0, {1\over n+1})$.
Then Kostya joined each pair
of marked points with a segment. Sasha drew a $1\times n$ rectangle
and joined with a segment each pair of integer points on its border.
As a result both the triangle and the rectangle are divided into polygons
by the segments drawn. Who has the greater number of polygons: Sasha or Kostya?
\rightline{({\it M.~Alekseyev}\/)}
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7. A $1\times 5n$ rectangle is partitioned into {\it tiles},
each of the tiles being either a separate $1\times 1$ square or a {\it
broken domino} consisting of two such squares separated by four squares
(not belonging to the domino). Prove that the
number of such partitions is a~perfect fifth power.
\rightline{({\it K.~Kokhas}\/)}
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8. In an acute triangle $ABC$ the points $C_m$, $A_m$, $B_m$ are the midpoints
of $AB$, $BC$, $CA$ respectively. Inside the triangle $ABC$ a point $P$
is chosen so that $\angle PCB=\angle B_mBC$ and $\angle PAB=\angle ABB_m$.
A line passing through $P$ and perpendicular to $AC$ meets the median
$BB_m$ at $E$. Prove that $E$ lies on the circumcircle of the triangle $A_mB_mC_m$.
\rightline{({\it K.~Ivanov}\/)}
\vfill\eject
\centerline{\it Junior league}
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\centerline{\tt First day}
\medskip
1. Arnim and Brentano have a little vase with 1500 candies on the table and
a huge sack with spare candies under the table. They play a game taking
turns, Arnim begins. At each move a~player can either eat 7 candies
or take 6 candies from under the table and add them to the vase. A player
cannot go under the table in two consecutive moves. A player is declared the winner
if he leaves the vase empty. In any other case, if a player cannot
make a move in his turn, the game is declared a tie. Is there a~winning
strategy for one of the players?
\rightline{({\it A.~Golovanov}\/)}
\smallskip
2. Given are integers $a$, $b$, $c$ and an odd prime $p$. Prove
that $p$ divides $x^2+y^2+ax+by+c$ for some integers $x$ and $y$.
\rightline{({\it A.~Golovanov}\/)}
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3. Bisectors of a right triangle $ABC$ with right angle $B$ meet at point $I$.
The perpendicular to $IC$ drawn from $B$ meets the line $IA$ at $D$;
the perpendicular to $IA$ drawn from $B$ meets the line $IC$ at $E$.
Prove that the circumcentre of the triangle $IDE$ lies on the line $AC$.
\rightline{({\it A.~Kuznetsov}\/)}
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4. Several {\it good\/} points, several {\it bad\/} points and several segments
are drawn in the plane. Each segment connects a good point and a bad one;
at most 100 segments begin at each point. We have paint of 200 colours.
One half of each segment is painted with one of these colours, and the other
half with another one. Is it always possible to do it so that every two
segments with common end are painted with four different colours?
\rightline{({\it M.~Qi, X.~Zhang}\/)}
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\centerline{\tt Second day}
\medskip
5. Each row of a $24\times 8$ table contains some permutation of
the numbers 1, 2, \dots, 8. In each column the numbers are multiplied.
What is the minimum possible sum of all the products?
\rightline{({\it C.~Wu}\/)}
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6. The city of Neverreturn has $N$ bus stops numbered 1, 2, \dots, $N$.
Each bus route is one-way and has only two stops, the beginning and the end.
The route network is such that departing from any stop one cannot return
to it using city buses.
When the mayor notices a route going from a stop with a greater number
to a stop with a lesser number, he orders to exchange the number plates
of its beginning and its end. Can the plate changing go on infinitely?
\rightline{({\it K.~Ivanov}\/)}
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7. $M$ is the midpoint of the side $AB$ in an equilateral triangle $ABC$.
The point $D$ on the side $BC$ is such that $BD : DC=3 : 1$.
On the line passing through $C$ and parallel to $MD$ there is a point $T$
inside the triangle $ABC$ such that $\angle CTA=150^\circ$.
Find the angle $MTD$.
\rightline{({\it K.~Ivanov}\/)}
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8. Eight poles stand along the road. A sparrow starts at the first pole
and once in a minute flies to a neighbouring pole. Let $a(n)$ be the number
of ways to reach the last pole in $2n+1$ flights (we assume $a(m)=0$ for $m<3$).
Prove that for all $n\geq 4$
$$
a(n)-7a(n-1)+15a(n-2)-10a(n-3)+a(n-4)=0.
$$
\rightline{({\it T.~Amdeberhan, F.~Petrov}\/)}
\end