2021 Fall AMC 12B Problems/Problem 13
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Problem
Let 
What is the value of
![\[\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?\]](http://latex.artofproblemsolving.com/5/c/b/5cb66360bc69b72e86eac3564c26bae7fd577bc9.png)
Solution
Plugging in 
, we get
![\[\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}=\frac{\sin \frac{6\pi}{11} \cdot \sin \frac{12\pi}{11} \cdot \sin \frac{18\pi}{11} \cdot \sin \frac{24\pi}{11} \cdot \sin \frac{30\pi}{11}}{\sin \frac{2\pi}{11} \cdot \sin \frac{4\pi}{11} \cdot \sin \frac{6\pi}{11} \cdot \sin \frac{8\pi}{11} \cdot \sin \frac{10\pi}{11}}.\]](http://latex.artofproblemsolving.com/f/f/8/ff84ea1918eb8039edc31877eb63ca11dfef1986.png)
Since 
and 
we get
![\[\frac{\sin \frac{6\pi}{11} \cdot \sin \frac{12\pi}{11} \cdot \sin \frac{18\pi}{11} \cdot \sin \frac{24\pi}{11} \cdot \sin \frac{30\pi}{11}}{\sin \frac{2\pi}{11} \cdot \sin \frac{4\pi}{11} \cdot \sin \frac{6\pi}{11} \cdot \sin \frac{8\pi}{11} \cdot \sin \frac{10\pi}{11}}=\frac{\sin \frac{6\pi}{11} \cdot \sin \frac{-10\pi}{11} \cdot \sin \frac{-4\pi}{11} \cdot \sin \frac{2\pi}{11} \cdot \sin \frac{8\pi}{11}}{\sin \frac{2\pi}{11} \cdot \sin \frac{4\pi}{11} \cdot \sin \frac{6\pi}{11} \cdot \sin \frac{8\pi}{11} \cdot \sin \frac{10\pi}{11}}=\boxed{\textbf{(E)}\ 1}.\]](http://latex.artofproblemsolving.com/e/7/e/e7e5d4583710682a595b107182f40d55b4cad8b7.png)
~kingofpineapplz ~Ziyao7294 (minor edit)
Solution 2
This question seems very similar to quadratic residues and reciprocities. It is like Gauss's lemma in number theory or Euler's criterion. Let 
where 
is an odd prime number and 
is any integer.
Then 
is the Legendre symbol 
. Legendre symbol is calculated using quadratic reciprocity which is 
. The Legendre symbol 
https://empslocal.ex.ac.uk/people/staff/rjchapma/courses/nt13/Eisenstein.pdf
~Lopkiloinm
Video Solution (Just 2 min!)
~Education, the Study of Everything
See Also
| 2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 12 |
Followed by Problem 14 |
| 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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