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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_25 / There is a unique sequence of integers a₁,a₂,⋯a₂₀₂₃ such that

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

There is a unique sequence of integers

a_1, a_2, \cdots a_{2023}

such that

\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}

whenever

\tan 2023x

is defined. What is $a_{2023}?$

Plain-text mathematical notation (without MathML)

There is a unique sequence of integers a₁,a₂,⋯a₂₀₂₃ such that
tan2023x=(a₁tanx+a₃tan³x+a₅tan⁵x+⋯+a₂₀₂₃tan²⁰²³x)/(1+a₂tan²x+a₄tan⁴x⋯+a₂₀₂₂tan²⁰²²x)whenever tan2023x is defined. What is $a_{2023}?$

Original LaTeX notation

There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that
\[\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}\]whenever $\tan 2023x$ is defined. What is $a_{2023}?$

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Source and history

upstream_id / Original problem ID: https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_25
task_id: 106053c5-e3ff-5636-94c4-8af2a68e3ed6
task_key: default--train--https~3a~2f~2fartofproblemsolving~2ecom~2fwiki~2findex~2ephp~2f2023~5fAMC~5f12A~5fProblems~2fProblem~5f25
task_revision_id: 2
config / split: default / train

Official source

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