300+ followers problem

Algebra Level 5

\[\large \dfrac{1}{x-y}+\dfrac{1}{y-z}+\dfrac{1}{x-z}\]

Let \(m\) be the minimum value of the above expression for reals \(x > y > z\) given \((x − y)(y − z)(x − z) = 300\).

Given that \(m\) can be expressed as \(\dfrac{1}{A}\sqrt[3]{\dfrac{B}{C}}\) where \(A, B, C\) are positive integers and \(\gcd(B,C)=1\), with \(B+C\) minimized.

Find \(A+B+C\).

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