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Omni-MATH / Find the largest real number λ with the following property: for any positive real numbers p,q,r,s there exists a complex number z=a+bi(a,b∈R) such that …

Problem

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problem

Find the largest real number

\lambda

with the following property: for any positive real numbers

p,q,r,s

there exists a complex number

z=a+bi

(

a,b\in \mathbb{R})

such that

|b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0.

Plain-text mathematical notation (without MathML)

Find the largest real number λ with the following property: for any positive real numbers p,q,r,s there exists a complex number z=a+bi(a,b∈R)  such that |b|≥λ|a| and (pz³+2qz²+2rz+s)⋅(qz³+2pz²+2sz+r)=0.

Original LaTeX notation

Find the largest real number $\lambda$ with the following property: for any positive real numbers $p,q,r,s$ there exists a complex number $z=a+bi$($a,b\in \mathbb{R})$  such that $$ |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0.$$

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Source and history

task_id: d374ab4f-f1fc-5f14-9e1d-d239ecda369f
task_key: test--d374ab4f-f1fc-5f14-9e1d-d239ecda369f
task_revision_id: 4
config / split: / test

Official source

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