Q:

Minimum value of $(a, b, c)$

If $a^2b^2+b^2c^2+c^2a^2-69abc=2016$, then, what can be said about the least value of $\min(a, b ,c)$, where $a, b, c\gt0$?
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 Issue 6 2016 of Mathematical Reflections.

A:

Let  $a,c \in \mathbb{R}_{>0}$. Assume $a = \epsilon > 0$ and $c  =  1$. This leads to the quadratic equation  in $b$
$$(1+\epsilon^2)b^2-69\epsilon b+\epsilon^2-2016 = 0,$$
with a solution given by
$$ b = \frac{69\epsilon +\sqrt{8064+12821\epsilon^2 -4\epsilon^4}}{2(1+\epsilon^2)} $$
Thus, $b \in \mathbb{R}_{>0}$ if and only if $8064+12821\epsilon^2 -4\epsilon^4 \geq 0$, which is true for $0 < \epsilon \leq 56.6205...$.
Thus, it is easy to conclude about  $\min (a,b,c)$.

