Problem of the Week #7: Monday Feb. 20th, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
a). A regular pentagram is a five-pointed star, as shown on the right. Notice it has a regular pentagon at its center shaded in gray. If these two figures are the same size, what is the ratio of the side lengths of the shaded pentagon to the large pentagon on the left?
b). There are two different regular octagrams, as shown in the middle and on the right here. Notice they each have a regular octagon in the middle shaded in gray. If all three figures are the same size, what is the ratio of side lengths of all three octagons?
c). For a regular polygon of n sides, find the number of unique regular polygrams and the side length ratio of all the n-sided regular polygons at their centers.
d). Share your own problem inspired by this scenario.
e). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.