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Omni-MATH / How many real triples (a,b,c) are there such that the polynomial p(x)=x⁴+ax³+bx²+ax+c has exactly three distinct roots, which are equal to tany,tan2y, and tan3y for some real y ?

Problem

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problem

How many real triples (a,b,c)(a, b, c) are there such that the polynomial p(x)=x4+ax3+bx2+ax+cp(x)=x^{4}+a x^{3}+b x^{2}+a x+c has exactly three distinct roots, which are equal to tany,tan2y\tan y, \tan 2 y, and tan3y\tan 3 y for some real yy ?
Plain-text mathematical notation (without MathML)
How many real triples (a,b,c) are there such that the polynomial p(x)=x⁴+ax³+bx²+ax+c has exactly three distinct roots, which are equal to tany,tan2y, and tan3y for some real y ?
Original LaTeX notation
How many real triples $(a, b, c)$ are there such that the polynomial $p(x)=x^{4}+a x^{3}+b x^{2}+a x+c$ has exactly three distinct roots, which are equal to $\tan y, \tan 2 y$, and $\tan 3 y$ for some real $y$ ?

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Source and history

task_id
c2ff4aa8-1490-5d70-9e64-51560158da26
task_key
test--c2ff4aa8-1490-5d70-9e64-51560158da26
task_revision_id
1
config / split
/ test

Official source

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