We know that:
[tex] x^2+y^2+z^2=110\\\\x^3+y^3+z^3=684\\\\xyz=210\\\\xy+yz+zx=107 [/tex]
so:
[tex] x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)\\\\684-3\cdot210=(x+y+z)\big(110-(xy+yz+zx)\big)\\\\684-630=(x+y+z)(110-107)\\\\54=(x+y+z)\cdot3\qquad|:3\\\\\boxed{x+y+z=18} [/tex]
