DK-GEORGMOHR 2006

The star shown is symmetric with respect to each of the six diagonals shown. All segments connecting the points $A_1, A_2, . . . , A_6$ with the centre of the star have the length $1$ , and all the angles at $B_1, B_2, . . . , B_6$ indicated in the figure are right angles. Calculate the area of the star.

Determine all sets of real numbers $(x,y,z)$ which fulfills $$\begin{cases} x + y =2 xy -z^2= 1\end{cases}$$

A natural number $n$ , which is at most $500$ , has the property that when one chooses at at random among the numbers $1, 2, 3,...,499, 500$ , then the probability is $\frac{1}{100}$ for $m$ to add up to $n$ . Determine the largest possible value of $n$ .

Of the numbers $1, 2,3,..,2006$ , ten different ones must be selected. Show that you can pick ten different numbers with a sum greater than $10039$ in more ways than you can select ten different numbers with a sum less than $10030$ .

We consider an acute triangle $ABC$ . The altitude from $A$ is $AD$ , the altitude from $D$ in triangle $ABD$ is $DE,$ and the altitude from $D$ in triangle $ACD$ is $DF$ . a) Prove that the triangles $ABC$ and $AF E$ are similar. b) Prove that the segment $EF$ and the corresponding segments constructed from the vertices $B$ and $C$ all have the same length.