Most Awaited Question

Algebra Level 5

If \(a\) and \(b\) are real numbers such that \(a^{2} + b^{2} = 1\) and it always satisfies

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

Find \(\left[ \frac{x}{z} \right]\), when the value of the above given inequality is least.


Here \(x\) and \(z\) are positive integers and \([•]\) denotes Greatest Integer Function.


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