DE-BUNDESWETTBEWERB 2012

Alex writes the sixteen digits $2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9$ side by side in any order and then places a colon somewhere between two digits, so that a division task arises. Can the result of this calculation be $2$ ?

Are there positive integers $a$ and $b$ such that both $a^2 + 4b$ and $b^2 + 4a$ are perfect squares?

An equilateral triangle $DCE$ is placed outside a square $ABCD$ . The center of this triangle is denoted as $M$ and the intersection of the straight line $AC$ and $BE$ with $S$ . Prove that the triangle $CMS$ is isosceles.

From the vertices of a regular 27-gon, seven are chosen arbitrarily. Prove that among these seven points there are three points that form an isosceles triangle or four points that form an isosceles trapezoid.

given a positive integer $n$ . the set $\{ 1,2,..,2n \}$ is partitioned into $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$ . find the value of : $\sum_{i=1}^{n}|a_i - b_i|$

On a round table, $n$ bowls are arranged in a circle. Anja walks around the table clockwise, placing marbles in the bowls according to the following rule: She places a marble in any first bowl, then goes one bowl further and puts a marble in there. Then she goes two shells before putting another marble, then she goes three shells, etc. If there is at least one marble in each shell, she stops. For which $n$ does this occur?

The incircle of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ in points $A_1, B_1$ and $C_1$ respectively. $C_1D$ is a diameter of the incircle. Finally, let $E$ be the intersection of the lines $B_1C_1$ and $A_1D$ . Prove that the segments $CE$ and $CB_1$ have equal length.

A rectangle with the side lengths $a$ and $b$ with $a <b$ should be placed in a right-angled coordinate system so that there is no point with integer coordinates in its interior or on its edge. Under what necessary and at the same time sufficient conditions for $a$ and $b$ is this possible?