Polynomial Centuries

Algebra Level 5

$$P(x)=x^{100}+a_{99}x^{99}+a_{98}x^{98}+...+a_{2}x^2+a_{1}x+1$$ is a polynomial function with all real and positive coefficients, that is, $$a_{1},a_{2},...,a_{99} \in R^{+}$$. It is known that all the roots of the equation $$P(x)=0$$ are real.

If $$\displaystyle \sum_{i=1}^{99} a_{i} \geq 2(2^{N}-1)$$ where $$N$$ is a positive integer, then find the maximum value of $$N$$.

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