Problem 22

Algebra Level 5

\[\large \left( \frac{x}{y}+\frac{y}{z}+\frac{z}{x} \right)^3 + \frac{525(xy+yz+xz)}{x^2+y^2+z^2}\]

If \(x,y,z\) are positive real numbers, the minimum value of the above expression can be expressed as \(\frac{m}{n}\) where \(m,n\) are positive integers that are relatively prime. Find \(m+n\).


This problem is in this Set.
Not an original problem, it's hard. But beautiful.

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