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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_12 / For complex number u=a+bi and v=c+di (where i=√(−1)), define the binary operation u⊗v=ac+bdi Suppose z is a complex number such that z⊗z=z²+40. What is |z|²?

Problem

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problem

For complex number

u = a+bi

and

v = c+di

(where

i=\sqrt{-1}

), define the binary operation

u \otimes v = ac + bdi

Suppose

z

is a complex number such that

z\otimes z = z^{2}+40

. What is

|z|^2

Plain-text mathematical notation (without MathML)

For complex number u=a+bi and v=c+di (where i=√(−1)), define the binary operation
u⊗v=ac+bdi
Suppose z is a complex number such that z⊗z=z²+40. What is |z|²?

Original LaTeX notation

For complex number $u = a+bi$ and $v = c+di$ (where $i=\sqrt{-1}$), define the binary operation
$u \otimes v = ac + bdi$
Suppose $z$ is a complex number such that $z\otimes z = z^{2}+40$. What is $|z|^2$?

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Source and history

upstream_id / Original problem ID: https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_12
task_id: 87c24820-e114-53eb-a4c8-75be264f5d7f
task_key: default--train--https~3a~2f~2fartofproblemsolving~2ecom~2fwiki~2findex~2ephp~2f2023~5fAMC~5f12B~5fProblems~2fProblem~5f12
task_revision_id: 2
config / split: default / train

Official source

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