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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