DANUBE 2008

In a triangle $ABC$ let $A_1$ be the midpoint of side $BC$ . Draw circles with centers $A, A1$ and radii $AA_1, BC$ respectively and let $A'A''$ be their common chord. Similarly denote the segments $B'B''$ and $C'C''$ . Show that lines $A'A'', B'B'''$ and $C'C''$ are concurrent.

On a semicircle centred at $O$ and with radius $1$ choose the respective points $A_1,A_2,...,A_{2n}$ , for $n \in N^*$ . The lenght of the projection of the vector $\overrightarrow {u}=\overrightarrow{OA_1} +\overrightarrow{OA_2}+...+\overrightarrow{OA_{2n}}$ on the diameter is an odd integer. Show that the projection of that vector on the diameter is at least $1$ .

Let $n\geq 2$ be a positive integer. Find the maximum number of segments with lenghts greater than $1,$ determined by $n$ points which lie on a closed disc with radius $1.$