Question 100
We call a set of three numbers "arithmetic" if one of its elements is the arithmetic mean of the other two. Similarly, we call a set of three numbers "harmonic" if one of its elements is the harmonic mean of the other two. How many three element subsets of the set
$$
\{z \mid -2011 \lt z \lt 2011\}
$$
of integers are both arithmetic and harmonic?
$$
\{z \mid -2011 \lt z \lt 2011\}
$$
of integers are both arithmetic and harmonic?