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OlympiadBench / 1613 / Determine all positive integers n satisfying the following condition: for…

Problem

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

question

Determine all positive integers nn satisfying the following condition: for every monic polynomial PP of degree at most nn with integer coefficients, there exists a positive integer k≤nk \leq n, and k+1k+1 distinct integers x1,x2,…,xk+1x_{1}, x_{2}, \ldots, x_{k+1} such that P(x1)+P(x2)+⋯+P(xk)=P(xk+1). P\left(x_{1}\right)+P\left(x_{2}\right)+\cdots+P\left(x_{k}\right)=P\left(x_{k+1}\right) . Note. A polynomial is monic if the coefficient of the highest power is one.
Plain-text mathematical notation (without MathML)
Determine all positive integers n satisfying the following condition: for every monic polynomial P of degree at most n with integer coefficients, there exists a positive integer k≤n, and k+1 distinct integers x₁,x₂,…,x_(k+1) such that



P(x₁)+P(x₂)+⋯+P(x_(k))=P(x_(k+1)).


Note. A polynomial is monic if the coefficient of the highest power is one.
Original LaTeX notation
Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k \leq n$, and $k+1$ distinct integers $x_{1}, x_{2}, \ldots, x_{k+1}$ such that



$$

P\left(x_{1}\right)+P\left(x_{2}\right)+\cdots+P\left(x_{k}\right)=P\left(x_{k+1}\right) .

$$


Note. A polynomial is monic if the coefficient of the highest power is one.

answer type

Numerical

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID
1613
task_id
a450231c-6472-5f4e-9bed-6208e8a06f8b
task_key
OE~5fTO~5fmaths~5fen~5fCOMP--train--1613
task_revision_id
3
config / split
OE_TO_maths_en_COMP / train

Official source

initial import