2008 iTest Problems/Problem 99
Given a convex, -sided polygon , form a -sided polygon by cutting off each corner of at the edges’ trisection points. In other words, is the polygon whose vertices are the edge trisection points of , connected in order around the boundary of . Let be an isosceles trapezoid with side lengths , and , and for each , let . This iterative clipping process approaches a limiting shape . If the difference of the areas of and is written as a fraction in lowest terms, calculate the number of positive integer factors of .
Let be the difference in the areas between and . Let our trapezoid be (and ); then without loss of generality construct diagonal .
Let be the trisection points on , respectively, that are closest to . Then the operation deletes . Since , and share common , we have by side ratio . Their areas are in the ratio .
Similarly, , and . Cutting along diagonal , we get the same result, so .
We now consider the effects of the second clipping. Without loss of generality consider what happens along the vertex of . Let be the trisection point along (again closest to ), and be the trisection point along . Now and , and . Using the definition of the area of a triangle, we see that . A similar clipping about gives ; around each clipped region in , we clip a new area . Generalizing, we have the recursion .
Then, . Hence,
Then has factors.