2013 AMC 12B Problem 16

An equiangular pentagon has all interior angles $108^\circ,$ so each point of the star is an isosceles triangle with base angles $72^\circ$ and apex $36^\circ.$ By the equal base angles, each point contributes two sides that are the same fixed multiple $c$ of the pentagon side it rests on. Summing over the five points, the star perimeter equals $2c\cdot(\text{pentagon perimeter}) = 2c,$ independent of the individual side lengths. So $s$ is constant, and the difference between its maximum and minimum values is $0.$ Thus, the correct answer is A.