Knife to me#2855
Birthday cake was very good: a tender coffee sponge cake with cream layers, covered in white chocolate — a miracle of culinary art. However, there was a problem. The cake was shaped like a prism whose base was a regular -gon—exactly matching the number of guests. But the unexpected arrival of Sasha complicated the situation. Now the cake must be divided into N+1 parts.
— Who here likes geometry? — Well, you studied at the math faculty too. — Incredible audacity. Since this is your fault, you get the knife. — Then I suggest we make vertical cuts. — Oh, Captain Obvious. — That's not all. Let's divide it evenly into N+1 parts so that all pieces have equal volume. — Alright. — That’s still not all. Let’s also split it so that the chocolate-covered surface area is equal for all pieces. — It would have been easier if I’d cut it myself.
We consider the projection of the cake onto the table. This forms a regular polygon centered at with one vertex at . Each required piece must be a convex polygon such that no three vertices of a piece are collinear. The cutting plan must fulfill these conditions regardless of the cake's height. Recall that only the exterior surface of the cake—the top and all side faces—is covered in chocolate.
The number (3 \leq N \leq 100) — the number of guests, not counting Sasha.
Print exactly N+1 blocks, each of which describes a single piece. The description of a piece begins with K_i (3 \leq K_i \leq 100), the number of vertices. Then, in the next K_i lines, list the vertices of the piece in anti-clockwise order, one vertex per line. An inaccuracy of up to 10^{-8} is allowed.
Esempio 1