Too long problem

Algebra Level 5

\[\large \frac{1}{x + y } + \frac{1}{y + z} + \frac{1}{z + x } = \frac{1}{2\left( x + y + z \right) }\]

If \(x,y\) and \(z\) are numbers such that the equation above is fulfilled, find the value of the expression below.

\[\frac{64\left( x + y + z \right) ^{6}- \left( x + y\right) ^{6}- \left( y + z\right) ^{6}- \left( z + x\right) ^{6}}{\left[ \left( x + y \right) ^{3}\left( y + z\right) ^{3}\right] + \left[ \left( y + z\right) ^{3}\left( z + x\right) ^{3}\right] + \left[ \left( z + x \right) ^{3}\left( x + y\right) ^{3}\right] } \]

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