\nopagenumbers
\input amssym.def
\input amssym.tex
\centerline{\bf ``TUYMAADA--2012''}
\medskip
\centerline{\it Senior league}
\medskip
\centerline{\tt First day}
\medskip
1. Tanya and Serezha take turns putting chips in empty squares of
a chessboard.
Tanya starts by putting a chip in an arbitrary square. At every next move
Serezha must put a chip in the column where Tanya put her last chip, and
Tanya must put a chip in the row where Serezha put his last chip.
The player that cannot move loses. Which of the players has a winning strategy?
\rightline{({\it A. Golovanov}\/)}
2. Quadratic trinomial $P(x)$ has two real roots and satisfies
the inequality $$P(x^3+x)\ge P(x^2+1)$$ for all $x$.
Find the sum of the roots of $P(x)$.
\rightline{({\it A. Golovanov, M. Ivanov, K. Kokhas}\/)}
3. A point $P$ is chosen inside the triangle $ABC$ so that
$$\angle PAB=\angle PCB={1\over 4}(\angle A+\angle C).$$ $BL$ is a
bisector of $ABC$. The line $PL$ meets the circumcircle of triangle $APC$ at point $Q$.
Prove that $QB$ is the bisector of $AQC$.
\rightline{({\it S. Berlov}\/)}
4. Let $p=4k+3$ be a prime, and ${m\over n}$ is irreducible fraction
such that $${1\over 0^2+1}+{1\over 1^2+1}+\dots+{1\over (p-1)^2+1}={m\over n}.$$
Prove that $p$ divides $2m-n$.
\rightline{({\it A. Golovanov}\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. Solve the equation
$${1\over n^2}-{3\over 2n^3}={1\over m^2}$$
in positive integers.
\rightline{({\it A. Golovanov}\/)}
6. Quadrilateral $ABCD$ is cyclic and circumscribed. Its incircle touches
its sides $AB$ and $CD$ at points $X$ and $Y$, respectively. The perpendiculars to $AB$ and $CD$
drawn at $A$ and $D$, respectively, meet at point $U$, those drawn at $X$
and $Y$ meet at point $V$, and, finally, those drawn at $B$ and $C$ meet at
point $W$. Prove that $U$, $V$, $W$ are collinear.
\rightline{({\it A. Golovanov}\/)}
7. Positive numbers $a$, $b$, $c$ satisfy $abc=1$. Prove that
$${1\over 2a^2+b^2+3}+{1\over 2b^2+c^2+3}+{1\over 2c^2+a^2+3}\leq {1\over 2}.$$
\rightline{({\it V. Aksenov}\/)}
8. Integers not divisible by 2012 are arranged on the arcs of an oriented
graph. We call {\it the weight of a vertex} the difference between the
sum of numbers on the arcs going to it and that on the arcs going from it.
It is known that the weight of each vertex is divisible by 2012. Prove
that non-zero integers with absolute values not exceeding 2012 can be
arranged on the arcs of this graph so that the weight of each vertex is zero.
\rightline{({\it W. Tutte}\/)}
\vfill\eject
\centerline{\it Junior league}
\medskip
\centerline{\tt First day}
\medskip
1. Tanya and Serezha take turns putting chips in empty squares of
a chessboard.
Tanya starts by putting a chip in an arbitrary square. At every next move
Serezha must put a chip in the column where Tanya put her last chip, and
Tanya must put a chip in the row where Serezha put his last chip.
The player that cannot move loses. Which of the players has a winning strategy?
\rightline{({\it A. Golovanov}\/)}
2. A rectangle $ABCD$ is given. Segment $DK$ is equal to $BD$ and lies
on the half-line $DC$. $M$ is the midpoint of $BK$. Prove that $AM$ is the
bisector of $BAC$.
\rightline{({\it S. Berlov}\/)}
3. Prove that $N^2$ arbitrary different positive integers ($N>10$) can be arranged
in $N\times N$ table so that all $2N$ sums in rows and columns are different.
\rightline{({\it S. Volchenkov}\/)}
4. Let $p=1601$ (a prime) and irreducible fraction ${m\over n}$
is the sum of those fractions
$${1\over 0^2+1},\quad{1\over 1^2+1},\quad\dots,\quad{1\over (p-1)^2+1},$$
whose denominators are not divisible by $p$.
Prove that $p$ divides $2m+n$.
\rightline{({\it A. Golovanov}\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. The vertices of a regular 2012-gon are denoted
$A_1$, $A_2$, \dots, $A_{2012}$ in some order.
It is known that if $k+l$ and $m+n$ leave the same remainder when divided
by 2012, then the chords $A_kA_l$ and $A_mA_n$ have no common points.
Vasya walks around the polygon and sees that the first two vertices
are denoted $A_1$ and $A_4$. How is the tenth vertex denoted?
\rightline{({\it A. Golovanov}\/)}
6. Solve the equation
$${1\over n^2}-{3\over 2n^3}={1\over m^2}$$
in positive integers.
\rightline{({\it A. Golovanov}\/)}
7. A circle lies in a quadrilateral with successive sides 3, 6, 5, 8.
Prove that its radius is less than 3.
\rightline{({\it K. Kokhas}\/)}
8. 25 little donkeys stand in a row; the rightmost of them is Eeyore.
Winnie the Pooh wants to give a balloon of one of the seven colours of
rainbow to each donkey so that successive donkeys receive balloons of
different colours and at least one ballon of each colour is given to somebody.
Eeyore wants to give to each of 24 remaining donkeys a pot of one
of six colours of rainbow (except red) so that at least one pot
of each colour is given to somebody (but successive donkeys can receive pots
of the same colour). Which of the friends has more ways to get his plan
implemented and how many times more?
\rightline{({\it F. Petrov}\/)}
\end