Question 8
Define a *regular $n$-pointed star* to be the union of $n$ line segments $\overline{P_1 P_2},$ $\overline{P_2 P_3},$ $\ldots,$ $\overline{P_n P_1}$ such that
• the points $P_1,$ $P_2,$ $\ldots,$ $P_n$ are coplanar and no three of them are collinear,
• each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint,
• all of the angles at $P_1,$ $P_2,$ $\ldots,$ $P_n$ are congruent,
• all of the $n$ line segments $\overline{P_1 P_2},$ $\overline{P_2 P_3},$ $\ldots,$ $\overline{P_n P_1}$ are congruent, and
• the path $P_1 P_2 \ldots P_n P_1$ turns counterclockwise at an angle of less than $180^\circ$ at each vertex.
There are no regular $3$-pointed, $4$-pointed, or $6$-pointed stars. All regular $5$-pointed stars are similar, but there are two non-similar regular $7$-pointed stars. How many non-similar regular $1000$-pointed stars are there?
• the points $P_1,$ $P_2,$ $\ldots,$ $P_n$ are coplanar and no three of them are collinear,
• each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint,
• all of the angles at $P_1,$ $P_2,$ $\ldots,$ $P_n$ are congruent,
• all of the $n$ line segments $\overline{P_1 P_2},$ $\overline{P_2 P_3},$ $\ldots,$ $\overline{P_n P_1}$ are congruent, and
• the path $P_1 P_2 \ldots P_n P_1$ turns counterclockwise at an angle of less than $180^\circ$ at each vertex.
There are no regular $3$-pointed, $4$-pointed, or $6$-pointed stars. All regular $5$-pointed stars are similar, but there are two non-similar regular $7$-pointed stars. How many non-similar regular $1000$-pointed stars are there?