Evidently the problem is to show this is impossible: $$\frac{z^3}{x^3+y^3+2xyz}=3$$$(x,y,z\in \mathbb N)$
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Why there isn't any solution in positive integers for $z^3 = 3(x^3 +y^3+2xyz)$?
How to prove if this equation provides an integral solution divisible by $3$?
A complete solution was given in comment by duje after my answer, elliptic curves.
ADDED: it seems one of the people asking has been wondering about similar problems for years:
http://www.science-bbs.com/121-math/19a7e45249b8ba6c.htm
I keep asking for the source, no joy. One possible reason for that is that this is part of some sort of contest, probably online but not necessarily in English. Oh, there seem to be a fair number of contests that are aimed at programmers but have explicit mathematical content.