The following problem looks very interesting to me and I cannot even guess a solution to it. It states that:
Suppose that $a,b,c$ are three natural numbers satisfying the inequality: $0\leq a^2 + b^2 - abc\leq c$. Show that $a^2 + b^2 - abc$ is a perfect square.
Cases like $a=b$ or $a=1$ can be handled very easily, but is there any general solution? Any help shall be greatly appreciated.