\nopagenumbers
\input amssym.def
\input amssym.tex
\centerline{\bf ``TUYMAADA--2013''}
\medskip
\centerline{\it Senior league}
\medskip
\centerline{\tt First day}
\medskip
1. 100 heaps of stones lie on a table. Two players make moves in turn.
At each move a player can remove any non-zero number of stones from at most
99 heaps. The player that cannot move loses.
Determine, for each initial position, which of the players, the first or
the second, has a winning strategy.
\rightline{({\it K. Kokhas}\/)}
\smallskip
2. Points $X$ and $Y$ inside the rhombus $ABCD$ are such that
$Y$ is inside the convex quadrilateral $BXDC$ and
$2\angle XBY=2\angle XDY=\angle ABC$. Prove that the lines $AX$ and $CY$
are parallel.
\rightline{({\it S. Berlov}\/)}
\smallskip
3. Vertices of a connected graph cannot be coloured with less than $n+1$
colours so that adjacent verices have different colours. Prove that
$n(n-1)/2$ edges can be removed from the graph so that it remains connected.
\rightline{({\it V. Dolnikov}\/)}
\smallskip
4. Prove that if $x$, $y$, $z$ are positive and $xyz=1$ then
$${x^3\over x^2+y}+{y^3\over y^2+z}+{z^3\over z^2+x}\geq {3\over 2}.$$
\rightline{({\it A. Golovanov}\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. Prove that every polynomial of fourth degree can be represented in the form $P(Q(x))+R(S(x))$, where $P$, $Q$, $R$, $S$ are quadratic trinomials.
\rightline{({\it A. Golovanov}\/)}
\smallskip
6. Solve the equation $p^2-pq-q^3=1$ in prime numbers.
\rightline{({\it A. Golovanov}\/)}
\smallskip
7. The points $A_1$, $A_2$, $A_3$, $A_4$ are vertices of a regular tetrahedron with edge 1.
The points $B_1$ and $B_2$ lie inside the figure bounded by the plane $A_1A_2A_3$
and the spheres of radius 1 with centres $A_1$, $A_2$, $A_3$.
Prove that $B_1B_2<\max(B_1A_1, B_1A_2, B_1A_3, B_1A_4)$.
\rightline{({\it A. Kupavsky}\/)}
\smallskip
8. Cards numbered from 1 to $2^n$ are distributed among $k$ children, $1\leq k\leq 2^n$, so that each child gets at least one card. Prove that the number of ways to do that is divisible by $2^{k-1}$ but not by $2^k$.
\rightline{({\it M. Ivanov}\/)}
\vfill\eject
\centerline{\it Junior league}
\medskip
\centerline{\tt First day}
\medskip
1. 100 heaps of stones lie on a table. Two players make moves in turn.
At each move a player can remove any non-zero number of stones from at most
99 heaps. The player that cannot move loses.
Determine, for each initial position, which of the players, the first or
the second, has a winning strategy.
\rightline{({\it K. Kokhas}\/)}
\smallskip
2. $ABCDEF$ is a convex hexagon such that $AC\parallel DF$,
$BD\parallel AE$ and $CE\parallel BF$. Prove that $AB^2+CD^2+EF^2=BC^2+DE^2+AF^2$.
\rightline{({\it N. Sedrakyan}\/)}
\smallskip
3. For every positive $a$ and $b$ prove the inequality
$$\sqrt{ab} \leq {1\over 3}\cdot\sqrt{a^2+b^2\over 2}+{2\over 3}\cdot{2\over{1\over a}+{1\over b}}.$$
\rightline{({\it A. Khrabrov}\/)}
\smallskip
4. Vertices of a connected graph cannot be coloured with less than $n+1$
colours so that adjacent verices have different colours. Prove that
$n(n-1)/2$ edges can be removed from the graph so that it remains connected.
\rightline{({\it V. Dolnikov}\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. Each face of a $7\times7\times7$ cube is divided into unit squares. What maximum number
of squares can be chosen so that no two chosen squares have a common point?
\rightline{({\it A. Golovanov}\/)}
\smallskip
6. Quadratic trinomials with positive leading coefficients are arranged in the squares of a $6\times$ table.
Their 108 coefficients are all integers from $-60$ to $47$ (each number is used once). Prove that at least in one column
the sum of all the trinomials has a real root.
\rightline{({\it K. Kokhas and F. Petrov}\/)}
\smallskip
7. Solve the equation $p^2-pq-q^3=1$ in prime numbers.
\rightline{({\it A. Golovanov}\/)}
\smallskip
8. The point $A_1$ on the perimeter of a convex quadrilateral $ABCD$ is such that the line $AA_1$
divides the quadrilateral into two parts of equal area. The points $B_1$, $C_1$, $D_1$ are defined similarly.
Prove that the area of the convex quadrilateral with vertices $A_1$, $B_1$, $C_1$, $D_1$ is greater than a~quarter of the area of $ABCD$.
\rightline{({\it L. Emelyanov}\/)}
\end