What is the strategy for solving Problem S395 in Mathematical Reflections?

[vidyarth]

If $a^2b^2+b^2c^2+c^2a^2-69abc=2016$, then, what can be said about the least value of $\min(a, b ,c)$?

This problem is unyielding to the major inequalities like AM-GM, Cauchy-Schwarz, etc. I also tried relating it to $x^3+y^3+z^3-3xyz=(x+y+z)(\sum_{cyc}x^2+\sum_{cyc}xy)$, but of no use. Any ideas. Thanks beforehand.

PS: This is problem S395 in Mathematical Reflections.

 

[Opalg]

If $a^2b^2+b^2c^2+c^2a^2-69abc=2016$, then, what can be said about the least value of $\min(a, b ,c)$?

This problem is unyielding to the major inequalities like AM-GM, Cauchy-Schwarz, etc. I also tried relating it to $x^3+y^3+z^3-3xyz=(x+y+z)(\sum_{cyc}x^2+\sum_{cyc}xy)$, but of no use. Any ideas. Thanks beforehand.

PS: This is problem S395 in Mathematical Reflections.

First, you need to include the information (given in the statement of Problem S395) that $a,b,c$ are positive integers. Without that information the problem does not make much sense.

Second, that section of Mathematical Reflections says that this problem is in a list whose deadline for submissions is January 15, 2017. So I wouldn't want to give away too many hints before then.

But just as a modest suggestion, I think that you might approach this problem along more naive lines than those that you suggest. If you want to minimise a positive integer then the smallest candidates are $1, 2, 3, \ldots $. So think about whether there is a possible solution with say $c=1$. If not , then how about $c=2,$ $c=3, \ldots$?