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2010 PUMaC Algebra A4/B6: nested radical

Source:

August 20, 2011
quadraticscalculusintegrationalgebraquadratic formula

Problem Statement

Define f(x)=x+x+x+x+x+…\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}. Find the smallest integer xx such that f(x)≥50xf(x)\ge50\sqrt{x}.
(Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)
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