Olympiad Problem 4

Algebra Level 5

The sequence \({x}_{n}\) is defined by \[x_1=\dfrac{1}{2},~~~ x_{k+1} = x_{k}^2 + x_k.\] If \[S=\dfrac{1}{{x}_{1}+1}+\dfrac{1}{{x}_{2}+1} + \dfrac{1}{{x}_{3}+1} +\cdots + \dfrac{1}{{x}_{100}+1}\] then find the value of \(\left\lfloor S\right\rfloor\).

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