#### Trigonometry IMO

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Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.

Solve the equation $\cos^2 x + \cos^2 2x +\cos^2 3x=1$ in $(0, 2\pi)$.

Prove for every positive integer $n$ and real number $x\ne \frac{k\pi}{2^t}$ where $t =0, 1, 2,\cdots$ and $k$ is an integer, the following relation always holds: $$\frac{1}{\sin 2x}+\frac{1}{\sin 4x} + \cdots +\frac{1}{\sin 2^nx}=\frac{1}{\tan x}-\frac{1}{\tan 2^nx}$$