If
![x,y,z](/media/m/b/7/2/b72c022e9d438802d328d34eb61bb4ba.png)
are real numbers satisfying relations
prove that
![x^{2n+1} + y^{2n+1} + z^{2n+1} = 1](/media/m/4/2/f/42f44d58282d9ebd24cf16ef8dae2885.png)
holds for all positive integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
.
%V0
If $x,y,z$ are real numbers satisfying relations
$$x+y+z = 1 \quad \textrm{and} \quad \arctan x + \arctan y + \arctan z = \frac{\pi}{4},$$
prove that $x^{2n+1} + y^{2n+1} + z^{2n+1} = 1$ holds for all positive integers $n$.