Prove that there are no integers $x, y, z$ such that $x+y+z=0$ and $1/x+1/y+1/z=0$ 
Prove that there are no integers $x, y, z$ such that $x+y+z=0$ and $\frac1x+\frac1y+\frac1z=0$

My thinking was that since the numbers are integers, then there can't be $2$ negative values that cancel out the positive or $2$ positive numbers to cancel the negative. One integer would have to cancel out the second, and the third wouldn't make the sum zero. Am I right?

We have that $$x+y=-z\quad\text{ and }\quad\frac1x+\frac1y=-\frac1z\\\implies\frac1{x+y}=\frac1x+\frac1y\implies (x+y)^2=xy\\\implies x^2+y^2=-xy\implies-\frac{x}y-\frac{y}x=1\tag1$$ for some $x,y,z\in\Bbb Z\setminus\{0\}$. Setting $r:=-\frac{x}y$ the last identity result in
$$r+r^{-1}=1\tag2$$
for some $r\in\Bbb Q\setminus\{0\}$, what is not possible because
$$r+r^{-1}=\frac{r^2+1}r=1\implies r^2-r+1=0\tag3$$
what doesnt have real solutions.
