problem 10

the question was
Let  be a finite set of at least two points in the plane. Assume that no three points of  are collinear. A windmill is a process that starts with a line  going through a single point . The line rotates clockwise about the pivot  until the first time that the line meets some other point belonging to . This point, , takes over as the new pivot, and the line now rotates clockwise about , until it next meets a point of . This process continues indefinitely. 
Show that we can choose a point  in  and a line  going through  such that the resulting windmill uses each point of  as a pivot infinitely many times.


now lets see its solution 

 This problem was very nice, creative and inspiring. Something a bit new which looks very nice. I will present a solution I found on AoPS, which I consider very interesting.

For any point , there is a line which cuts the remaining points in two groups, which differ with at most  point from one another. This kind of a line works for the problem. Why? Imagine how such a line creates the windmill: it touches one point, then one side gains one point, and then this side loses the pivot, leaving the situation as before, i.e. with sides of the line differing by at most one point.

Pick now one point and a line which has the above property. At a given moment, the windmill will be parallel to this line, and by uniqueness, will coincide with this line. This reasoning shows that any point will be pivot in a  rotation of the windmill.