qgis - Cut off overlapping polygon areas at their cutting edge

I have a polygon layer with many circle polygons in it, which are overlapping each other.

I now would like to cut off the overlapping areas at their cutting edge:

I use QGIS 2.2 and couldn't find a tool to do it, all vector tools are only working if there are two different layers to cut (there is now way to convert it to different layers because I have over 100 polygons, which are overlapping each other different times.)