DE-BUNDESWETTBEWERB 2008

Fedja used matches to put down the equally long sides of a parallelogram whose vertices are not on a common line. He figures out that exactly 7 or 9 matches, respectively, fit into the diagonals. How many matches compose the parallelogram's perimeter?

Represent the number $2008$ as a sum of natural number such that the addition of the reciprocals of the summands yield 1.

In a planar coordinate system we got four pieces on positions with coordinates. You can make a move according to the following rule: You can move a piece to a new position if there is one of the other pieces in the middle of the old and new position. Initially the four pieces have positions $\{(0,0),(0,1),(1,0),(1,1)\}$ . Given a finite number of moves can you yield the configuration $\{(0,0), (1,1), (3,0), (2, - 1)\}$ ?

Let the positive integers $a,b,c$ chosen such that the quotients $\frac{bc}{b+c},$ $\frac{ca}{c+a}$ and $\frac{ab}{a+b}$ are integers. Prove that $a,b,c$ have a common divisor greater than 1.