Difference between revisions of "2024 AIME II Problems/Problem 15"
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| − | + | ==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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
