MathDB

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} \mbox{and} H(a,b)=\frac{2ab}{a+b}=\frac{2}{\frac{1}{a}+\frac{1}{b}}\mbox{,} respectively, where

H(a,b)

is not defined for some

a

b

Problem Statement