2008 iTest Problems/Problem 99
Problem
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
.
Solution
Let be the difference in the areas between
and
. Let our trapezoid be
(and
); then without loss of generality construct diagonal
.
![[asy] pathpen = linewidth(0.7);pen d = linetype("4 4")+linewidth(0.7); pair A=(0,0),B=(5,12),C=(8,12),D=(13,0),A1=B/3,A2=D/3; D(MP("A",A)--MP("B",B,N)--MP("C",C,N)--MP("D",D)--cycle); D(B--D); D(MP("A_1",A1,NW) -- MP("A_2",A2)); D(D+2(C-D)/3 -- B+2(C-B)/3); D(4B/9 -- (A+D)/3+2*((A+B)/3 - (A+D)/3)/3, d); D(4D/9 -- (A+D)/3 + ((A+B)/3 - (A+D)/3)/3, d); [/asy]](http://latex.artofproblemsolving.com/f/3/0/f30c6ba339233428c01745f5f17d417b372f0006.png)
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
.
Similarily, , 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,
![$P_{10} - P_{\infty} = 96\left(\left(\frac 29\right)^{10} + \left(\frac 29\right)^{11} + \cdots \right) = \left(\frac 29\right)^{10}\left(\frac{1}{1-2/9}\right) = \frac{2^{15}}{3^{17} \cdot 7}.$](http://latex.artofproblemsolving.com/1/7/3/17394ffc48d5d6d6bb0bc98263ce824189a85146.png)
Then has
factors.