8-0. Once, King Shahryar said to Scheherazade: "Here is a paper circle with 1001 points on its boundary. Each night, you must cut the figure you have along a straight line containing any two of the marked points, keeping only one fragment and discarding the other. And make sure that the figure you keep is not a polygon, but a figure from which a polygon can be obtained by further cutting." On which night will Scheherazade, even with her most skillful actions, no longer be able to fulfill Shahryar's condition?
# Answer: 1999

Solution. First, let's describe the strategy that Scheherazade will use to meet Shahryar's conditions for 1998 nights. Initially, she will cut along the lines connecting adjacent vertices and discard the smaller part (let's call such a part a segment). She will do this for 1000 nights. After this, a 1001-gon with a segment attached to one of its sides will remain. After this, she will cut off triangles. After each such cut, the number of vertices decreases by 1 (and a segment is still attached to one of the sides), and after another 998 moves, a triangle with a segment will remain, after which there will be nothing left to cut according to the rules.

Similarly, it can be proven that Scheherazade has no strategy for more moves. With each move, Scheherazade discards either a segment or a vertex (or both). Eventually, a segment and at least three vertices must remain, which means that it is impossible to make more than 1998 cuts.