AMC 12 2006a Problem 23

Given a finite sequence $S=(a_1,a_2,\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence

$\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)$

of $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\le m\le n-1$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and let $S=(1,x,x^2,\ldots ,x^{100})$. If $A^{100}(S)=(1/2^{50})$, then what is $x$?

$\mathrm{(A) \ } 1-\frac{\sqrt{2}}{2} \mathrm{(B) \ } \sqrt{2}-1 \mathrm{(C) \ } \frac{1}{2} \mathrm{(D) \ } 2-\sqrt{2} \mathrm{(E) \ } \frac{\sqrt{2}}{2}$