TITLE: What is the motivation behind the solution of this olympiad problem?
QUESTION [5 upvotes]: Is there a particular reason why solutions in arithmetic progression were sought for? Can you think of an alternate solution?
Note: This is a problem from the Indian National Mathematical Olympiad 2001.

REPLY [6 votes]: The righthand side suggests naming the differences. We might as well assume that $x\le y\le z$ and set $a=y-x$ and $b=z-y$. The equation then becomes
$$3y^2+a^2+b^2+2(b-a)y=ab(a+b)\;,$$
and at that point it's easy to see that setting $a=b$ would simplify matters greatly. At this point it could still turn out, of course, that the simplification is useless, because the additional requirement kills off too many solutions, but it's so much easier to work with that it should definitely be investigated, and it turns out to be fruitful.