Tripping triples

Algebra Level 5

An ordered triple of pairwise distinct real numbers (a,b,c)(a,b,c) is called tripping if it satisfies {3a+b=a3,3b+c=b3,3c+a=c3. \begin{cases} 3a+b=a^3,\\ 3b+c=b^3,\\ 3c+a=c^3.\\ \end{cases}

How many tripping triples are there?

Details and assumptions

A set of values is called pairwise distinct if no two of them are equal. For example, the set {1,2,2} \{1, 2, 2\} is not pairwise distinct, because the last 2 values are the same.

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