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Difference between revisions of "2014 AMC 12B Problems/Problem 25"

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Now we use product-to-sum identities to get:
 
Now we use product-to-sum identities to get:
  
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<cmath>\frac{1}{2} \left(\cos{2x + \left( \frac{2014\pi^2}{x} \right) } + \cos{2x + \left( \frac{2014\pi^2}{x} \right) }) \right) = 1</cmath>
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<cmath>\frac{1}{2} \left(\cos{2x + \left( \frac{2014\pi^2}{x} \right) } + \cos{2x + \left( \frac{2014\pi^2}{x} \right) } \right) = 1</cmath>
  
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<cmath>\cos{2x + \left( \frac{2014\pi^2}{x} \right) } + \cos{2x + \left( \frac{2014\pi^2}{x} \right) } \right) = 1</cmath>
+
<cmath>\cos{2x + \left( \frac{2014\pi^2}{x} \right) } + \cos{2x + \left( \frac{2014\pi^2}{x} \right) } = 1</cmath>
  
 
Notice that for any <math>\theta</math>, <math>\max{\cos \theta} = 1</math>. This is achieved when <math>\theta = 0</math>, or equivalently  
 
Notice that for any <math>\theta</math>, <math>\max{\cos \theta} = 1</math>. This is achieved when <math>\theta = 0</math>, or equivalently  

Revision as of 16:18, 26 September 2022

Problem

Find the sum of all the positive solutions of

$2\cos2x \left(\cos2x - \cos{\left( \frac{2014\pi^2}{x} \right) } \right) = \cos4x - 1$

$\textbf{(A)}\ \pi \qquad\textbf{(B)}\ 810\pi \qquad\textbf{(C)}\ 1008\pi \qquad\textbf{(D)}\ 1080 \pi \qquad\textbf{(E)}\ 1800\pi$

Solution 1

Rewrite $\cos{4x} - 1$ as $2\cos^2{2x} - 2$. Now let $a = \cos{2x}$, and let $b = \cos{\left( \frac{2014\pi^2}{x} \right) }$. We have: \[2a(a - b) = 2a^2 - 2\]

Therefore, \[ab = 1\].

Notice that either $a = 1$ and $b = 1$ or $a = -1$ and $b = -1$. For the first case, $a = 1$ only when $x = k\pi$ and $k$ is an integer. $b = 1$ when $\frac{2014\pi^2}{k\pi}$ is an even multiple of $\pi$, and since $2014 = 2*19*53$, $b =1$ only when $k$ is an odd divisor of $2014$. This gives us these possible values for $x$: \[x= \pi, 19\pi, 53\pi, 1007\pi\] For the case where $a = -1$, $\cos{2x} = -1$, so $x = \frac{m\pi}{2}$, where m is odd. $\frac{2014\pi^2}{\frac{m\pi}{2}}$ must also be an odd multiple of $\pi$ in order for $b$ to equal $-1$, so $\frac{4028}{m}$ must be odd. We can quickly see that dividing an even number by an odd number will never yield an odd number, so there are no possible values for $m$, and therefore no cases where $a = -1$ and $b = -1$. Therefore, the sum of all our possible values for $x$ is \[\pi + 19\pi + 53\pi + 1007\pi = \boxed{\textbf{(D)}\ 1080 \pi}\]

Solution 2

Very similar to the solution above, re-write the expression using $\cos 4x = 2 \cos^2 2x - 1$:

\[2\cos2x \left(\cos2x - \cos{\left( \frac{2014\pi^2}{x} \right) } \right) = 2 \cos^2 2x - 2\]

Now, expand the LHS and cancel terms:

\[2 \cos^2 2x - 2 \cos 2x \cos{\left( \frac{2014\pi^2}{x} \right) } = 2 \cos^2 2x - 2\]

\[\cos 2x \cos{\left( \frac{2014\pi^2}{x} \right) } = 1\]

Now we use product-to-sum identities to get:

\[\frac{1}{2} \left(\cos{2x + \left( \frac{2014\pi^2}{x} \right) } + \cos{2x + \left( \frac{2014\pi^2}{x} \right) } \right) = 1\]

\[\cos{2x + \left( \frac{2014\pi^2}{x} \right) } + \cos{2x + \left( \frac{2014\pi^2}{x} \right) } = 1\]

Notice that for any $\theta$, $\max{\cos \theta} = 1$. This is achieved when $\theta = 0$, or equivalently

\[2x - \frac{2014\pi^2}{x} \equiv 0 \mod 2\pi\]

We can cleverly assume $x=c\pi$ for some real $c$. Then, we must have

\[2c - \frac{2014}{c} \equiv 0 \mod 2\]

In order for this to be satisfied, $\frac{2014}{c}$ must be an even integer. Factoring $2014 = 2 \cdot 19 \cdot 53$, we see that our only positive valid $c = 19, 53, 1007$. Our answer is just $\pi(19+53+1007) = 1080\pi \implies \textbf{(D)}$.


-FIREDRAGONMATH16

See also

2014 AMC 12B (Problems • Answer Key • Resources)
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Problem 24 		Followed by
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All AMC 12 Problems and Solutions

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