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Fraction/Polynomial Inequality - Max Value

Source: 2015 AIME 2 Problem 8

March 26, 2015

inequalities

Problem Statement

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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