{"status": "success", "data": {"description_md": "Given a finite sequence $S=(a_1,a_2,\\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence \n\n$\\left(\\frac{a_1+a_2}{2},\\frac{a_2+a_3}{2},\\ldots ,\\frac{a_{n-1}+a_n}{2}\\right)$
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 $x$?\n\n$\\mathrm{(A) \\ } 1-\\frac{\\sqrt{2}}{2}\\qquad \\mathrm{(B) \\ } \\sqrt{2}-1\\qquad \\mathrm{(C) \\ } \\frac{1}{2}\\qquad \\mathrm{(D) \\ } 2-\\sqrt{2}\\qquad \\mathrm{(E) \\ } \\frac{\\sqrt{2}}{2}$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "

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

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2006 AMC 12A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/06_amc12A_p24", "prev": "/problem/06_amc12A_p22"}}