DK-GEORGMOHR 2002

An interior point in a rectangle is connected by line segments to the midpoints of its four sides. Thus four domains (polygons) with the areas $a, b, c$ and $d$ appear (see the figure). Prove that $a + c = b + d$ .

Prove that for any integer $n$ greater than $5$ , a square can be divided into $n$ squares.

Two positive integers have the sum $2002$ . Can $2002$ divide their product?

In triangle $ABC$ we have $\angle C = 90^o$ and $AC = BC$ . Furthermore $M$ is an interior pont in the triangle so that $MC = 1 , MA = 2$ and $MB =\sqrt2$ . Determine $AB$

Homer Grog has written the numbers $1, 3, 4, 5, 7, 9, 11, 13, 15,17$ , one number on each note. He arranges the bills in a circle and tries to get the largest sum $S$ of the numbers of three consecutive bills to be the least possible. What is the smallest value $S$ can assume?