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max of (xyz)/(1 + 5x)(4x + 3y)(5y + 6z)(z + 18) when x,y,z>0

Source: Singapore Open Math Olympiad 2003 2nd Round p2 SMO

April 2, 2020

maxinequalitiesalgebra

Problem Statement

Find the maximum value of

\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}

x, y

and

z

range over the set of all positive real numbers. Justify your answer.