DANUBE 2008 Problem 3

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 $1$ .