Find all integer pairs $(a,b)$ such that $a^2+ab+b^2=219^2$, without technology.
My colleague showed me this question. He said it is from the AIME (American Invitational Mathematics Examination).
I regarded the equation as a quadratic in $b$, so $$b=\frac{-a\pm\sqrt{438^2-3a^2}}{2}$$ so $438^2-3a^2$ must be a square number. Without technology, I have not found a nice way to find all possible values of $a$.
EDIT: I don't know if this helps, but if we ignore the bit about "without technology", then using excel we can find that the solutions are:
$\pm(0, 219), \pm(51, 189), \pm(51, -240), \pm(189,-240), (219, -219)$ and their inverses.
EDIT2: My colleague, to my annoyance, just told me that the whole question was this: A triangle has side lengths $a, b, 219$ where $a$ and $b$ are integers. The angle opposite the side of length $219$ is $120°$. Find $a$ and $b$.
I apologize for posting an incomplete, possibly unsolvable, question and thus wasting readers' time. In future, when posting questions that I get from other people, I will be more careful to avoid the X-Y problem.
Having said that, I am still interested in whether the original version of the question is solvable.