Difference between revisions of "2024 AIME II Problems/Problem 15"
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We use inclusion-exclusion for this. There are 30 valid rectangles contained in <math>A_1A_5A_7A_{11}</math>, as well as 30 in <math>A_2A_4A_8A_{10}</math>. However, the 9 rectangles contained in <math>B_1B_2B_3B_4</math> have been counted twice, so we subtract 9 and we have 51 rectangles in the diagram. Multiplying by 3, we get 153 rectangles for Case 1. | We use inclusion-exclusion for this. There are 30 valid rectangles contained in <math>A_1A_5A_7A_{11}</math>, as well as 30 in <math>A_2A_4A_8A_{10}</math>. However, the 9 rectangles contained in <math>B_1B_2B_3B_4</math> have been counted twice, so we subtract 9 and we have 51 rectangles in the diagram. Multiplying by 3, we get 153 rectangles for Case 1. | ||
− | For Case 2, we have this diagram. | + | For Case 2, we have this diagram. To be honest, you can count the rectangles here in whatever way you like. |
<asy> | <asy> | ||
real r = pi/6; | real r = pi/6; |
Revision as of 19:33, 8 February 2024
Problem
Find the number of rectangles that can be formed inside a fixed regular dodecagon 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.
Solution 1
By Furaken
There are two kinds of such rectangles: those whose sides are parallel to some edges of the regular 12-gon (Case 1, and those whose sides are not (Case 2).
For Case 1, WLOG assume that the rectangle's sides are horizontal and vertical (don't forget to multiply by 3 at the end of Case 1). Then the rectangle's sides coincide with these segments as shown in the diagram.
We use inclusion-exclusion for this. There are 30 valid rectangles contained in
, as well as 30 in
. However, the 9 rectangles contained in
have been counted twice, so we subtract 9 and we have 51 rectangles in the diagram. Multiplying by 3, we get 153 rectangles for Case 1.
For Case 2, we have this diagram. To be honest, you can count the rectangles here in whatever way you like.
There are 36 rectangles contained within
, and 18 that use points outside
. So we get a total of
rectangles for Case 2.
Adding the two cases together, we get the answer .
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 |
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