AT-OEMO-FED1 2011 Problem 3

A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two.

How many three-element subsets of the set of integers $\left\{z\in\mathbb{Z}\mid -2011<z<2011\right\}$ are arithmetic and harmonic?

(Remark: The arithmetic mean $A(a,b)$ and the harmonic mean $H(a,b)$ are defined as $$A(a,b)=\frac{a+b}{2} \text{and} H(a,b)=\frac{2ab}{a+b}=\frac{2}{\frac{1}{a}+\frac{1}{b}}\text{,}$$ respectively, where $H(a,b)$ is not defined for some $a$ , $b$ .)