Question 14
Let $A_1A_2\cdots A_{101}$ be a regular 101-gon, and color every vertex red or blue. Let $N$ be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the 101-gon, both the vertices with acute angles have the same color, and the vertex with obtuse angle have different color.
(1) Find the largest possible value of $N$.
(2) Find the number of ways to color the vertices such that maximum $N$ is achieved. (Two colorings are different if for some $A_i$ the colors are different on the two coloring schemes)
Supplemental illustration (AI-generated; gray vertices, if present, have unspecified colors):
(1) Find the largest possible value of $N$.
(2) Find the number of ways to color the vertices such that maximum $N$ is achieved. (Two colorings are different if for some $A_i$ the colors are different on the two coloring schemes)
Supplemental illustration (AI-generated; gray vertices, if present, have unspecified colors):