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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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