AIME 2015 Problem 8

Algebra Level 5

Let \(a\) and \(b\) be positive integers satisfying \(\dfrac{ab+1}{a+b} < \frac{3}{2}\). The maximum possible value of \(\dfrac{a^3b^3+1}{a^3+b^3}\) is \(\dfrac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q\).


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