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Part of 2011 Federal Competition For Advanced Students, Part 1

Problems(1)

Arithmetic and harmonic three-element Sets

Source: Austrian Federal Competition for Advanced Students 2011, #3

5/15/2011
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 {z∈Z∣−2011<z<2011}\left\{z\in\mathbb{Z}\mid -2011<z<2011\right\} are arithmetic and harmonic?
(Remark: The arithmetic mean A(a,b)A(a,b) and the harmonic mean H(a,b)H(a,b) are defined as A(a,b)=\frac{a+b}{2} \mbox{and}  H(a,b)=\frac{2ab}{a+b}=\frac{2}{\frac{1}{a}+\frac{1}{b}}\mbox{,} respectively, where H(a,b)H(a,b) is not defined for some aa, bb.)
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