The Answer Is Not 3

Algebra Level 5

Consider all ordered triples of positive real numbers $(a, b, c)$ such that $a^2 + b^2 + c^2 = \frac{5}{2} (ab + bc + ca ) .$ To 3 decimal places, what is the minimum value of

$\frac { a+b+c} { \sqrt[3]{abc} } ?$

It should be clear that $a = b = c$ doesn't satisfy the given condition, which is why the answer is not 3.
You will likely need to use a calculator to evaluate the final expression.