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?

Hint: if $x+y+z=0$, then
$$
\begin{align}
\frac1x+\frac1y+\frac1z
&=\frac1x+\frac1y-\frac1{x+y}\\
&=\frac{(x+y)^2+x^2+y^2}{2xy(x+y)}\\
&=-\frac{x^2+y^2+z^2}{2xyz}
\end{align}
$$
