Generalization of an IMO Problem!

For discussing Olympiad Level Algebra (and Inequality) problems
Nirjhor
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Joined:Thu Aug 29, 2013 11:21 pm
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Generalization of an IMO Problem!

Unread post by Nirjhor » Tue Mar 04, 2014 1:14 pm

1. Prove that: \[\cos \dfrac\pi7 - \cos\dfrac {2\pi}{7}+\cos\dfrac{3\pi}{7}=\dfrac{1}{2}\]
2. Generalization (and a useful hint for the above): Prove that, $\forall ~k\geq 3$ such that $k$ is an odd integer: \[\sum_{n=1}^{(k-1)/2}\cos\dfrac{(2n-1)\pi}{k}=\dfrac{1}{2}\]
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.


Revive the IMO marathon.

Nirjhor
Posts:136
Joined:Thu Aug 29, 2013 11:21 pm
Location:Varies.

Re: Generalization of an IMO Problem!

Unread post by Nirjhor » Thu Mar 06, 2014 10:45 pm

Hint: $\cos\dfrac{5\pi}{7}=\cos\left(\pi-\dfrac{2\pi}{7}\right)=-\cos\dfrac{2\pi}{7}$
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.


Revive the IMO marathon.

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