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OlympiadBench / 1620 / Let n≥2 be an integer, and let f be a 4n-variable polynomial…

Problem

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

question

Let

n \geqslant 2

be an integer, and let

f

be a

4 n

-variable polynomial with real coefficients. Assume that, for any

2 n

points

\left(x_{1}, y_{1}\right), \ldots,\left(x_{2 n}, y_{2 n}\right)

in the plane,

f\left(x_{1}, y_{1}, \ldots, x_{2 n}, y_{2 n}\right)=0

if and only if the points form the vertices of a regular

2 n

-gon in some order, or are all equal. Determine the smallest possible degree of

f

Plain-text mathematical notation (without MathML)

Let n≥2 be an integer, and let f be a 4n-variable polynomial with real coefficients. Assume that, for any 2n points (x₁,y₁),…,(x_(2n),y_(2n)) in the plane, f(x₁,y₁,…,x_(2n),y_(2n))=0 if and only if the points form the vertices of a regular 2n-gon in some order, or are all equal.



Determine the smallest possible degree of f.

Original LaTeX notation

Let $n \geqslant 2$ be an integer, and let $f$ be a $4 n$-variable polynomial with real coefficients. Assume that, for any $2 n$ points $\left(x_{1}, y_{1}\right), \ldots,\left(x_{2 n}, y_{2 n}\right)$ in the plane, $f\left(x_{1}, y_{1}, \ldots, x_{2 n}, y_{2 n}\right)=0$ if and only if the points form the vertices of a regular $2 n$-gon in some order, or are all equal.



Determine the smallest possible degree of $f$.

answer type

Expression

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1620
task_id: 9a9418d8-258a-5c80-ba4b-a65f3e8fb960
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1620
task_revision_id: 3
config / split: OE_TO_maths_en_COMP / train

Official source

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