Proto P#3687
Consider a regular -gon centered at the origin , with one of its vertices located at . Your task is to divide this polygon into exactly p-1 convex polygons, ensuring that each of these smaller polygons has the same perimeter.
You will be given an integer where 3 \leq p \leq 23.
Provide p-1 sections, each detailing a polygon. For each polygon, include the following description: the integer , representing the number of vertices (3 \leq n \leq 100), followed by lines, each containing two real numbers that specify the coordinates of the vertices in a counterclockwise sequence. Ensure that no three vertices are collinear, and that each polygon is convex.
Each polygon must occupy a distinct area, and all polygons should have identical perimeters. The solution will be deemed correct if all specified measurements (such as intersection areas, perimeters, and vertex angles) are accurate within an absolute or relative error of 10^{-8}. Please present the coordinates with at least decimal places.
Example 1