Inspired by Daniel Liu

Algebra Level 5

Consider all sets of real numbers \(x, y, z \) which satisfy the equation

\[ xy^2 + yz^2 + zx^2 = 3 \]

Let the infimum value of \( \frac{ x ^2 + y^2 } { z^2 } + \frac{ y^2 + z^2 } { x^2 } + \frac{ z^2 + x^2 } { y^2 } \) be denoted as \( n \).

For how many sets of ordered triples \( ( x, y, z ) \) which satisfy the initial equation, also satisfy

\[ \frac{ x ^2 + y^2 } { z^2 } + \frac{ y^2 + z^2 } { x^2 } + \frac{ z^2 + x^2 } { y^2 } = n ? \]

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