2021 Fall AMC 12B Problems/Problem 15
Contents
Problem
Three identical square sheets of paper each with side length are stacked on top of each other. The middle sheet is rotated clockwise about its center and the top sheet is rotated clockwise about its center, resulting in the -sided polygon shown in the figure below. The area of this polygon can be expressed in the form , where , , and are positive integers, and is not divisible by the square of any prime. What is ?
Solution 1
The -sided polygon is made out of shapes like . Then , and , so . Then ; therefore . Thus and the required area is . Finally . ~lopkiloinm
Solution 2
As shown in Image:2021_AMC_12B_(Nov)_Problem_15,_sol.png, all 12 vertices of three squares form a regular dodecagon (12-gon). Denote by the center of this dodecagon.
Hence, .
Because the length of a side of a square is 6, .
Hence, .
We notice that . Hence, .
Therefore, the area of the region that three squares cover is
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
Solution 3 (complex number & coordinate geometry)
set A = 3+3i , A' , B' rotate 30 degree from A, B
A'= A
line A'B'
intersect with line y=3 at point , then length ,
use shoelace or = 1/2 * AE * AB/2 = 1/2 * * 3
total area = 24 * = = 108 - 36 the answer is .
Video Solution (Just 4 min!)
~Education, the Study of Everything
Video Solution by TheBeautyofMath
~IceMatrix
See Also
2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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