The Inequality Inspired by Joel Tan

Algebra Level 5

Let \(x, y, z\geq 0\) be reals such that \(x+y+z=1\).

Find the maximum possible value of

\[x (x+y)^5(y+z)^4(x+z)^4\]

The answer can be written as \(\dfrac {a^i}{b^jc^k}\) for positive integers \(a, b, c, i, j, k\), where \(a, b, c\) are as small as possible. Find \(a+b+c+i+j+k\).



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