2006 AMC 12A Problems/Problem 23

Problem

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

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

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 $x$ ?

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

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2006 AMC 12A Problems/Problem 23

Problem

Solution

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