INMO 2006 Problem 3

Let $X$ denote the set of all triples $(a, b, c)$ of integers. Define a function $f: X \rightarrow X$ by

$$f(a, b, c)=(a+b+c, a b+b c+c a, a b c)$$

Find all triples $(a, b, c)$ in $X$ such that $f(f(a, b, c))=(a, b, c)$.