MathDB

Points

A_1, A_2, ..., A_{10}

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

A_1, A_2, ... , A_9

and the tenth grasshopper was on arc

A_9A_{10}A_1

. Is it necessarily true that this grasshopper was exactly at point

A_{10}

Problem Statement