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: \[\sum_{n=1}^{(k-1)/2}\cos\dfrac{(2n-1)\pi}{k}=\dfrac{1}{2}\]
Generalization of an IMO Problem!
- 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.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
Re: Generalization of an IMO Problem!
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.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.