DE-BUNDESWETTBEWERB 2011

Ten bowls are in a circle. They will go clockwise - starting somewhere filled with $1, 2, 3, ..., 9$ or $10$ marbles. You can have two choices in every move . Add a marble to neighboring shells or from two neighboring shells - if both of them are not empty - remove one marble each. Can you achieve that after finally many moves in each bowl exactly $2011$ marbles lying?

$16$ children are sitting at a round table. After the break, they sit down again on table. They find that each child is either sitting on its original [lace or in one of the two neighboring places. How many seating arrangements are possible in this way after the break?

Let $a$ and $b$ be positive integers. As is known, the division of of $a \cdot b$ with $a + b$ determines integers $q$ and $r$ uniquely such that $a \cdot b = q (a + b) + r$ and $0 \le r <a + b$ . Find all pairs $(a, b)$ for which $q^2 + r = 2011$ .

Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than $180^o$ .

Proove that if for a positive integer $n$ , both $3n + 1$ and $10n + 1$ are perfect squares , then $29n + 11$ is not a prime number.

When preparing for a competition with more than two participating teams two of them play against each other at most once. When looking at the game plan it turns out: (1) If two teams play against each other, there are no more team playing against both of them. (2) If two teams do not play against each other, then there is always exactly two other teams playing against them both. Prove that all teams play the same number of games.

Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$ , sides $BD$ and $AC$ have the same length $b$ , side $AB$ has length $c_1$ and the side $CD$ has length $c_2$ . There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$ .