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problem about function

Source: China south east mathematical Olympiad 2006 problem1

July 4, 2013
functionalgebra unsolvedalgebra

Problem Statement

Suppose a>b>0a>b>0, f(x)=2(a+b)x+2ab4x+a+bf(x)=\dfrac{2(a+b)x+2ab}{4x+a+b}. Show that there exists an unique positive number xx, such that f(x)=(a13+b132)3f(x)=\left(\dfrac{a^{\frac{1}{3}}+b^{\frac{1}{3}}}{2} \right)^3.
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