Points A1,A2,...,A10 are marked on a circle clockwise. It is known that these points can be divided into pairs of points symmetric with respect to the centre of the circle. Initially at each marked point there was a grasshopper. Every minute one of the grasshoppers jumps over its neighbour along the circle so that the resulting distance between them doesn't change. It is not allowed to jump over any other grasshopper and to land at a point already occupied. It occurred that at some moment nine grasshoppers were found at points A1,A2,...,A9 and the tenth grasshopper was on arc A9A10A1. Is it necessarily true that this grasshopper was exactly at point A10? combinatoricscombinatorial geometry