226. In a regular 1976-gon, the midpoints of all sides and the midpoints of all diagonals are marked. What is the greatest number of marked points that can lie on one circle?
226. Answer: 1976. All marked points, except for the center $O$ of the 1976-gon, lie 1976 each on 987 circles with center $O$. Any other circle $\gamma$ intersects each of these 987 circles at two points; besides these intersection points, there can be only one more marked point on $\gamma$: $O$. Therefore, there are no more than $987 \cdot 2 + 1 = 1975$ marked points on such a circle $\gamma$.