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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_22 / Let c be a real number, and let z₁ and z₂ be the two complex numbers satisfying the equation z²−cz+10=0. Points z₁, z₂, (1)/(z₁), and (1)/(z₂) are the vertices of (convex) quadrila…

Problem

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problem

Let

c

be a real number, and let

z_1

and

z_2

be the two complex numbers satisfying the equation

z^2 - cz + 10 = 0

. Points

z_1

z_2

\frac{1}{z_1}

, and

\frac{1}{z_2}

are the vertices of (convex) quadrilateral

\mathcal{Q}

in the complex plane. When the area of

\mathcal{Q}

obtains its maximum possible value, let

c=\sqrt{m}

. what is the value of m

Plain-text mathematical notation (without MathML)

Let c be a real number, and let z₁ and z₂ be the two complex numbers satisfying the equation
z²−cz+10=0. Points z₁, z₂, (1)/(z₁), and (1)/(z₂) are the vertices of (convex) quadrilateral Q in the complex plane. When the area of Q obtains its maximum possible value, let c=√(m). what is the value of m

Original LaTeX notation

Let $c$ be a real number, and let $z_1$ and $z_2$ be the two complex numbers satisfying the equation
$z^2 - cz + 10 = 0$. Points $z_1$, $z_2$, $\frac{1}{z_1}$, and $\frac{1}{z_2}$ are the vertices of (convex) quadrilateral $\mathcal{Q}$ in the complex plane. When the area of $\mathcal{Q}$ obtains its maximum possible value, let $c=\sqrt{m}$. what is the value of m

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Source and history

upstream_id / Original problem ID: https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_22
task_id: edfe1acb-e472-5939-b702-3d4056000410
task_key: default--train--https~3a~2f~2fartofproblemsolving~2ecom~2fwiki~2findex~2ephp~2f2022~5fAMC~5f12A~5fProblems~2fProblem~5f22
task_revision_id: 2
config / split: default / train

Official source

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