Most Awaited Question

Algebra Level 4

$$a$$ and $$b$$ are real numbers such that $$a^{2} + b^{2} = 1,$$ and always satisfy

$\frac{1}{1 + a^{2}}+ \frac{1}{1 + b^{2}} + \frac{1}{1 + ab} \geq \frac{x}{1 + \frac{(a+b)^{2}}{z}}.$

Find $$\left\lfloor \frac{x}{z} \right\rfloor$$ when the value of the LHS of the above inequality is the least.


Details and Assumptions:

• $$x$$ and $$z$$ are positive integers.
• $$\lfloor \cdot \rfloor$$ denotes the greatest integer function.
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