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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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