This is an interesting problem I came across:
In a regular 2014-gon, all mid-points of all sides and diagonals are marked. What is the maximal number of such points that belong to a same circle?
I don't know how to tackle it. Of course, some programming could be applied. Is there some solution without programming (although I would accept Mathematica or Matlab code as a valid answer)? And, is 2014 the answer, or some larger number (most likely multiple of 2014)?? I couldn't intuitively answer that.
What about any regular polygon? Does, lets say, primality of number of sides affect the answer in such case?