Find all triples $\left(x,y,z\right)$of natural nonzero numbers,such that:
$$\arctan\frac{1}{x}+\arctan\frac{1}{y}+\arctan\frac{1}{z}=\frac{\pi}{4}.$$
Using complex numbers and relationship b/w angle addition when multiplying complex numbers, we essentially want to find sols to $xy+xz+yz+x+y+z=xyz+1$, but how to find the number of solutions?
Without loss of generality $x\le y\le z$, so$$\arctan\frac1x\ge\frac{\pi}{12}\implies x\le\left\lfloor\cot\frac{\pi}{12}\right\rfloor=3.$$Similarly, $z\ge4$. Now try the cases $x=1,\,x=2,\,x=3$ separately.
If $x=1$ then $y+z=0$, which doesn't work.
If $x=2$ then $(y-3)(z-3)=10$, which has solutions $(y,\,z)\in\{(4,\,13),\,(5,\,8)\}$.
If $x=3$ then $(y-2)(z-2)=5$, so $y=3,\,z=7$.