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Algebra Level 4

The real numbers $a , b , c , x , y , z$ satisfy $a\geq b \geq c > 0$ and $x \geq y \geq z >0$

The minimum value of

${\dfrac{(ax)^{2}}{(by + cz)(bz + cy)}} + {\dfrac{(by)^{2}}{(cz + ax)(cx + az)}} + {\dfrac{(cz)^{2}}{(ax + by)(ay + bx)}}$

can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are positive coprime integers. Find $50(m + n)$

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