10.38 In a regular 1976-gon, the midpoints of all sides and the midpoints of all diagonals are marked. How many of the marked points can lie on the same circle?
[Solution] In a regular 1976-gon, there are 987 diagonals of different lengths. The longest diagonals total 988, and the midpoints of each of these diagonals are the center $O$ of the regular 1976-gon. The midpoints of the remaining diagonals of each length lie on a circle centered at $O$. The midpoints of the sides of the regular 1976-gon also lie on a circle centered at $O$. In this way, we obtain 987 concentric circles, each with 1976 marked points.

Any circle $S$, which intersects each of these 987 circles at most at two points, even if point $O$ is also on $S$, has at most 1975 marked points on circle $S$. Therefore, the maximum number of points is 1976.