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.

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