Sum of fractions, again

Algebra Level 5

\[\large\frac{x(yz+1)^2}{z^2(zx+1)}+\frac{y(zx+1)^2}{x^2(xy+1)}+\frac{z(xy+1)^2}{y^2(yz+1)}\]

Let \(x,y,z>0\) satisfy \(x+y+z\leq\frac{3}{2}\). If the minimum value of the expression above can be written as \(\dfrac{a}{b}\) which \(a\) and \(b\) are coprime positive integers, calculate the product \(ab\).


This problem was taken from Lam Son Highschool grade 10 entrance test for mathematically specialized students
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