Equality or Inequality, That is the question!

Algebra Level 5

How many \(\textbf {unordered}\) triplets \((x,y,z)\) , subject to constraints, \((x^4-2x^3)_{\text{cyclic}}\leq0\) , satisfy the system of equations:

\[\left\{\begin{array}{l}(4x^2-8x+3)\sqrt{(2x-x^2)}+1=y\\ (8y-4y^2-3)\sqrt{(2y-y^2)}+1=z\\ (4z^2-8z+3)\sqrt{(2z-z^2)}+1=x\end{array}\right.\]

Details and Assumptions:

\(\bullet\) \((\cdots)_{\text{cyclic}}\) means that the relation holds true for all individual \((x,y,z)\)

\(\bullet\) An \(\textbf{unordered}\) triplet means that \((1,2,3)\) is the same as \((3,2,1)\) or \((1,3,2)\).

It is inspired by one of my favourite problems by Daniel Liu.

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