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- Feb 14, 2012

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Find positive $a$ and $b$ so that $\dfrac{a+b}{a(\tan a+\tan b)}=2015$.

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- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,894

-----

Find positive $a$ and $b$ so that $\dfrac{a+b}{a(\tan a+\tan b)}=2015$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- #2

- Feb 14, 2012

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Solution from kaliprasad :

If we take $a=\frac{\pi}{4}$ then we have $\tan a = 1$

If we choose $\tan b = 0$ then we have $a + b = \frac{2015\pi}{4}$ or $b = \frac{1007\pi}{2}$ or $\tan b$ infinite which is a contradiction.

If we choose $\tan b = 1$ then we have $a + b = \frac{4030\pi}{4}$ or $b = \frac{4029\pi}{4}$ or $\tan b$ 1 which is consistent.

So $a=\frac{\pi}{4},b=\frac{4029\pi}{4}$ is a solution.

Alternative solution:

Substituting these into the original equation and simplifying, we have

$\dfrac{2m+2n+1}{4m+1}=2015$

Observe that $n=5036$ and $m=1$ leading to a possible set of solution, i.e. $(a,\,b)=\left(\dfrac{5\pi}{4},\,\dfrac{20145\pi}{4}\right)$.

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