Problem 22

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

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 $m+n$ .

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Inspiration .

This problem is in this Set .

Not an original problem, it's hard. But beautiful.