Most Awaited Question

$a$ 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.

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Details and Assumptions:

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