Grasshoppers jumping on circle Ten points $A_1,\dots,A_{10}$ are marked in clockwise order on a circle, so that $A_iA_{i+5}$ forms a diameter for all $1\leq i\leq 5$. Initially, a grasshopper is at each point. Every minute, one grasshopper jumps over another grasshopper so that * the distance between them before and after the jump is the same. * the grasshopper did not jump over any grasshopper other than the one it jumped over, and it did not land on the same point as any other grasshopper. After some time, there is a grasshopper on each of the points $A_1,\dots,A_9$, and the last grasshopper is on the arc $A_9A_{10}A_1$. Is it necessary that it is exactly on point $A_{10}$?

This is not an answer but a demonstration that for the number of points $n = 4$ it is NOT true that the last grasshopper must be at the $A_n$ position.

First a (not very good) drawing: ![Grasshoppers for n = 4]( There are 4 points on the circle, A1, A2, A3 and A4. Points A1 and A3 are at the top and bottom of the circle. Points A2 and A4 are respectively, slightly below the rightmost point and slightly above the leftmost point of the circle.

Grasshopper at A4 makes it's first jump, arriving at the red position. It then makes the second jump to the orange position. It then makes the third jump to the green position.

So at this time, there are grasshoppers at A1, A2 and A3, the jumping rules have been followed, and yet the grasshopper from A4 is in the A3-A4-A1 arc without it being in the original A4 position.

For $n=4$ it is therefore not necessary that $A_n$ is occupied by a grasshopper.