1. Solve the equation $\frac{1}{x+y+z}=\overline{0, x y z}$ (here $x, y$ and $z-$ are some digits).
3.II.1. The equation $\overline{x y z} \cdot(x+y+z)=1000$. If $x=0$, then $y+z50$, but there are no two-digit divisors of 1000. The number 1000 has five natural three-digit divisors: $100, 125, 200, 250$, and 500. Direct verification shows that the equation is satisfied only by the triplet $(1; 2; 5)$.

Answer: $\{(1; 2; 5)\}$.