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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_16 / Consider the set of complex numbers z satisfying |1+z+z²|=4. The maximum value of the imaginary part of z can be written in the form (√(m))/(n), where m and n are relatively prime …

Problem

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problem

Consider the set of complex numbers

z

satisfying

|1+z+z^{2}|=4

. The maximum value of the imaginary part of

z

can be written in the form

\tfrac{\sqrt{m}}{n}

, where

m

and

n

are relatively prime positive integers. What is

m+n

Plain-text mathematical notation (without MathML)

Consider the set of complex numbers z satisfying |1+z+z²|=4. The maximum value of the imaginary part of z can be written in the form (√(m))/(n), where m and n are relatively prime positive integers. What is m+n?

Original LaTeX notation

Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

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Source and history

upstream_id / Original problem ID: https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_16
task_id: e7024150-ee17-580b-9393-bf7a7cdf7a85
task_key: default--train--https~3a~2f~2fartofproblemsolving~2ecom~2fwiki~2findex~2ephp~2f2023~5fAMC~5f12A~5fProblems~2fProblem~5f16
task_revision_id: 2
config / split: default / train

Official source

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