<s>[INST] 3 complex-variable equation 
Moderator Note: This is a current contest question on Brilliant.org.

$x,y,z$ are complex numbers satisfying
$$
\begin{align}
x+y+z & =1\\ 
x^2+y^2+z^2 & =2\\
x^3+y^3+z^3 & =3
\end{align}
$$
The value of $x^4+y^4+z^4$ can be expressed as $\dfrac ab$, where $a$ and $b$ are positive co-prime integers. What is the value of $a+b$ ?
 [/INST] HINT:
$(x+y+z)^2=(x^2+y^2+z^2)+2(xy+yz+zx)$
$\implies 2(xy+yz+zx)=(x+y+z)^2-(x^2+y^2+z^2)=1^2-2=-1$
Again, $x^3+y^3+z^3-3xyz=(x+y+z)\{(x+y+z)^3-3(xy+yz+zx)\}$
So, we have found $x+y+z,xy+yz+zx, xyz$
So, we can form the cubic equation whose roots are $x,y,z$
Can you take it from here?

Alternatively,
$$x^4+y^4+z^4=(x^2+y^2+z^2)^2-2(x^2y^2+y^2+z^2+z^2x^2)$$
$$=4-2(x^2y^2+y^2+z^2+z^2x^2)$$
Now, $x^2y^2+y^2+z^2+z^2x^2=(xy+yz+zx)^2-2(xy\cdot yz+yz\cdot zx+zx\cdot xy)$
$=(xy+yz+zx)^2-2xyz(x+y+z)$
</s>