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Cube roots in an equation

Source: Germany 2009 - Problem 1

December 31, 2014
geometry3D geometryfunctioncalculusalgebra unsolvedalgebra

Problem Statement

Find all non-negative real numbers aa such that the equation 1+x3+1−x3=a \sqrt[3]{1+x}+\sqrt[3]{1-x}=a has at least one real solution xx with 0≤x≤10 \leq x \leq 1. For all such aa, what is xx?
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