Algebraic System of Equations

Adding the three equations: $x + y + z - (\frac{1}{x} + \frac{1}{y} + \frac{1}{z}) = y + z + x \Rightarrow \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0$. So A is correct.

Squaring both sides of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0$ gives $\sum \frac{1}{x^2} + 2\sum \frac{1}{xy} = 0$. Also squaring original equations $x^2 + \frac{1}{x^2} - 2 = y^2$, summing gives $\sum \frac{1}{x^2} = 6$. So B is incorrect. Then $6 + 2\sum \frac{1}{xy} = 0 \Rightarrow \sum \frac{1}{xy} = -3$. So C is correct.

Thus, A and C only.