Polynomial Centuries

Algebra Level 5

P(x)=x100+a99x99+a98x98+...+a2x2+a1x+1P(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, a1,a2,...,a99R+a_{1},a_{2},...,a_{99} \in R^{+}. It is known that all the roots of the equation P(x)=0P(x)=0 are real.

If i=199ai2(2N1) \displaystyle \sum_{i=1}^{99} a_{i} \geq 2(2^{N}-1) where NN is a positive integer, then find the maximum value of NN.

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