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A cubic eqn

Source: INMO 1997 Problem 6

October 6, 2005
functionabsolute valuealgebra unsolvedalgebra

Problem Statement

Suppose aa and bb are two positive real numbers such that the roots of the cubic equation x3−ax+b=0x^3 - ax + b = 0 are all real. If α\alpha is a root of this cubic with minimal absolute value, prove that ba<α<3b2a. \dfrac{b}{a} < \alpha < \dfrac{3b}{2a}.
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