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Difference between revisions of "2021 Fall AMC 12B Problems/Problem 13"

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Let <math>c=\frac{2\pi}{p}</math> where <math>p</math> is an odd prime number and <math>n</math> is a relatively prime number to <math>p</math>.
 
Let <math>c=\frac{2\pi}{p}</math> where <math>p</math> is an odd prime number and <math>n</math> is a relatively prime number to <math>p</math>.
  
−
Then <math>\dfrac{\sin(nc)\sin(2nc)\ldots\sin(\frac{p-1}{2}nc)}{\sin(c)\sin(2c)\ldots\sin(\frac{p-1}{2}c)}</math> is the Legendre symbol of <math>\left(\frac{n}{p}\right)</math> as famous German Number Theoretician Ferdinand Gotthold Max Eisenstein used to prove the Legendre symbol. Any sensible person who researched Legendre symbols on wikipedia right before the competition would have gotten it in 5 seconds. Legendre symbol is calculated using quadratic reciprocity. <math>\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}</math>
+
Then <math>\dfrac{\sin(nc)\sin(2nc)\ldots\sin(\frac{p-1}{2}nc)}{\sin(c)\sin(2c)\ldots\sin(\frac{p-1}{2}c)}</math> is the Legendre symbol of <math>\left(\frac{n}{p}\right)</math> as famous German Number Theoretician Ferdinand Gotthold Max Eisenstein used to prove the Legendre symbol. Any sensible person who researched Legendre symbols on wikipedia right before the competition would have gotten it in 5 seconds. Legendre symbol is calculated using quadratic reciprocity which is <math>\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}</math>. The Legendre symbol <math>\left(\frac{3}{11}\right)=(-1)\left(\frac{11}{3}\right)=(-1)\left(\frac{-1}{3}\right)=(-1)(-1)=1</math>
 
https://en.wikipedia.org/wiki/Legendre_symbol#cite_ref-7
 
https://en.wikipedia.org/wiki/Legendre_symbol#cite_ref-7
  

Revision as of 21:29, 16 November 2022

Problem

Let $c = \frac{2\pi}{11}.$ What is the value of \[\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?\]

$\textbf{(A)}\ {-}1 \qquad\textbf{(B)}\ {-}\frac{\sqrt{11}}{5} \qquad\textbf{(C)}\ \frac{\sqrt{11}}{5} \qquad\textbf{(D)}\ \frac{10}{11} \qquad\textbf{(E)}\ 1$

Solution

Plugging in $c$, we get \[\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}=\frac{\sin \frac{6\pi}{11} \cdot \sin \frac{12\pi}{11} \cdot \sin \frac{18\pi}{11} \cdot \sin \frac{24\pi}{11} \cdot \sin \frac{30\pi}{11}}{\sin \frac{2\pi}{11} \cdot \sin \frac{4\pi}{11} \cdot \sin \frac{6\pi}{11} \cdot \sin \frac{8\pi}{11} \cdot \sin \frac{10\pi}{11}}.\] Since $\sin(x+2\pi)=\sin(x),$ $\sin(2\pi-x)=\sin(-x),$ and $\sin(-x)=-\sin(x),$ we get \[\frac{\sin \frac{6\pi}{11} \cdot \sin \frac{12\pi}{11} \cdot \sin \frac{18\pi}{11} \cdot \sin \frac{24\pi}{11} \cdot \sin \frac{30\pi}{11}}{\sin \frac{2\pi}{11} \cdot \sin \frac{4\pi}{11} \cdot \sin \frac{6\pi}{11} \cdot \sin \frac{8\pi}{11} \cdot \sin \frac{10\pi}{11}}=\frac{\sin \frac{6\pi}{11} \cdot \sin \frac{-10\pi}{11} \cdot \sin \frac{-4\pi}{11} \cdot \sin \frac{2\pi}{11} \cdot \sin \frac{8\pi}{11}}{\sin \frac{2\pi}{11} \cdot \sin \frac{4\pi}{11} \cdot \sin \frac{6\pi}{11} \cdot \sin \frac{8\pi}{11} \cdot \sin \frac{10\pi}{11}}=\boxed{\textbf{(E)}\ 1}.\]

~kingofpineapplz ~Ziyao7294 (minor edit)

Solution 2

Let $c=\frac{2\pi}{p}$ where $p$ is an odd prime number and $n$ is a relatively prime number to $p$.

Then $\dfrac{\sin(nc)\sin(2nc)\ldots\sin(\frac{p-1}{2}nc)}{\sin(c)\sin(2c)\ldots\sin(\frac{p-1}{2}c)}$ is the Legendre symbol of $\left(\frac{n}{p}\right)$ as famous German Number Theoretician Ferdinand Gotthold Max Eisenstein used to prove the Legendre symbol. Any sensible person who researched Legendre symbols on wikipedia right before the competition would have gotten it in 5 seconds. Legendre symbol is calculated using quadratic reciprocity which is $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}$. The Legendre symbol $\left(\frac{3}{11}\right)=(-1)\left(\frac{11}{3}\right)=(-1)\left(\frac{-1}{3}\right)=(-1)(-1)=1$ https://en.wikipedia.org/wiki/Legendre_symbol#cite_ref-7

~Lopkiloinm

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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