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Algebra Level 3

aa and bb are real numbers such that a2+b2=1,a^{2} + b^{2} = 1, and always satisfy

11+a2+11+b2+11+abx1+(a+b)2z. \frac{1}{1 + a^{2}}+ \frac{1}{1 + b^{2}} + \frac{1}{1 + ab} \geq \frac{x}{1 + \frac{(a+b)^{2}}{z}}.

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


Details and Assumptions:

  • xx and zz are positive integers.
  • \lfloor \cdot \rfloor denotes the greatest integer function.
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