2024 AIME II Problems/Problem 15: Difference between revisions
Mrmustache (talk | contribs) No edit summary |
R00tsofunity (talk | contribs) No edit summary |
||
| Line 1: | Line 1: | ||
==Problem== | |||
Find the number of rectangles that can be formed from a regular dodecagon such that 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(60); | |||
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(A3--A2--A9--A8--cycle, mediumgray); | |||
draw(A5--A12); | |||
dot(0.365*A4); | |||
dot(0.365*A1); | |||
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> | |||
==See also== | |||
{{AIME box|year=2024|n=II|num-b=14|after=Last Problem}} | |||
[[Category:Intermediate Combinatorics Problems]] | |||
{{MAA Notice}} | |||
Revision as of 17:01, 8 February 2024
Problem
Find the number of rectangles that can be formed from a regular dodecagon such that 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(60); 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(A3--A2--A9--A8--cycle, mediumgray); draw(A5--A12); dot(0.365*A4); dot(0.365*A1); 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]](http://latex.artofproblemsolving.com/7/4/8/748fdb39a6662dc1c6bacaea068baa0ce4471201.png)
See also
| 2024 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Last Problem | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America. 
