2002 I #5: Level 2
Source: 2024 AIME II Problem 15
February 8, 2024
AMCAIMEAIME IIgeometryrectangle
Problem Statement
Find the number of rectangles that can be formed inside a fixed regular dodecagon (-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles.[asy]
unitsize(40);
real r = pi/6;
pair A1 = (cos(r),sin(r));
pair A2 = (cos(2r),sin(2r));
pair A3 = (cos(3r),sin(3r));
pair A4 = (cos(4r),sin(4r));
pair A5 = (cos(5r),sin(5r));
pair A6 = (cos(6r),sin(6r));
pair A7 = (cos(7r),sin(7r));
pair A8 = (cos(8r),sin(8r));
pair A9 = (cos(9r),sin(9r));
pair A10 = (cos(10r),sin(10r));
pair A11 = (cos(11r),sin(11r));
pair A12 = (cos(12r),sin(12r));
draw(A1--A2--A3--A4--A5--A6--A7--A8--A9--A10--A11--A12--cycle);
filldraw(A2--A1--A8--A7--cycle, mediumgray, linewidth(1.2));
draw(A4--A11);
draw(0.365*A3--0.365*A12, linewidth(1.2));
dot(A1);
dot(A2);
dot(A3);
dot(A4);
dot(A5);
dot(A6);
dot(A7);
dot(A8);
dot(A9);
dot(A10);
dot(A11);
dot(A12);
[/asy]