# Welcome 2016! Part 23

Let $$a$$ and $$b$$ be positive integers satisfying $$\dfrac{ab+1}{a+b} < \dfrac{3}{2}$$.

The maximum possible value of $$\dfrac{a^3b^3+1}{a^3+b^3}$$ is $$\dfrac{p}{q}$$, where $$p$$ and $$q$$ are coprime positive integers. Find $$p+q$$.

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