<p>Let <img alt="$a$" height="10" src="https://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png" width="11" /> and <img alt="$b$" height="14" src="https://latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png" width="10" /> be positive integers satisfying <img alt="$\frac{ab+1}{a+b} < \frac{3}{2}$" height="39" src="https://latex.artofproblemsolving.com/1/a/a/1aa6efb035f7bb99e25c3c65752ba4530cc77d26.png" width="88" />. The maximum possible value of <img alt="$\frac{a^3b^3+1}{a^3+b^3}$" height="40" src="https://latex.artofproblemsolving.com/9/6/7/9677407e8f6e20fd32ad74d487fdd2d272cf1259.png" width="66" /> is <img alt="$\frac{p}{q}$" height="36" src="https://latex.artofproblemsolving.com/7/3/4/7347a1d5586e82c832f61ea799d89945ed2837fc.png" width="11" />, where <img alt="$p$" height="11" src="https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png" width="10" /> and <img alt="$q$" height="11" src="https://latex.artofproblemsolving.com/0/6/1/0615acc3725de21025457e7d6f7694dab8e2f758.png" width="8" /> are relatively prime positive integers. Find <img alt="$p+q$" height="14" src="https://latex.artofproblemsolving.com/1/7/0/170200756543596efeb46fca1cf29a617e8c9192.png" width="41" />.</p>

Question No: 390

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

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Tags: Algebra

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