Reciprocal of Square Roots Sum

Algebra Level 4

\[S = \frac 1{\sqrt 1 + \sqrt 3 } + \frac 1{\sqrt 5 + \sqrt 7 } + \frac 1{\sqrt 9 + \sqrt {11} } + \cdots + \frac 1{\sqrt {9997} + \sqrt {9999}} \]

Find \(\lfloor S \rfloor \).

Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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