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Min of (|b|+|c|)/a

Source: 2021 Taiwan APMO Preliminary First Round

November 21, 2020
algebra

Problem Statement

Let the three roots of x3+ax+1=0x^3+ax+1=0 be α,β,γ\alpha,\beta,\gamma where aa is a positive real number. Let the three roots of x3+bx2+cx−1=0x^3+bx^2+cx-1=0 be αβ,βγ,γα\frac{\alpha}{\beta},\frac{\beta}{\gamma},\frac{\gamma}{\alpha}. Find the minimum value of ∣b∣+∣c∣a\dfrac{|b|+|c|}{a}.
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