Olympiad Problem 6

Algebra Level 4

$\large (ab+bc+ca) \left (\frac 1{(a+b)^2} + \frac 1{(b+c)^2}+\frac 1{(c+a)^2} \right)$

For $$a, b, c$$ are positive reals, if the infimum (minimum) value of the expression above can be expressed as $$\frac{ x}{y}$$ for positive coprime integers $$x$$ and $$y$$, find the value of $$x - y$$.

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