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OlympiadBench / 1970 / Consider all polynomials P(x) with real coefficients that have the following…

Problem

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

answer type

Interval

is multiple answer

false

language

English

question

Consider all polynomials

P(x)

with real coefficients that have the following property: for any two real numbers

x

and

y

one has

$$
\left|y^{2}-P(x)\right| \leqslant 2|x| \text { if and only if }\left|x^{2}-P(y)\right| \leqslant 2|y|
\tag{1}
$$

Determine all possible values of

P(0)

Plain-text mathematical notation (without MathML)

Consider all polynomials P(x) with real coefficients that have the following property: for any two real numbers x and y one has

$$
\left|y^{2}-P(x)\right| \leqslant 2|x| \text { if and only if }\left|x^{2}-P(y)\right| \leqslant 2|y|
\tag{1}
$$

Determine all possible values of P(0).

Original LaTeX notation

Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has

$$
\left|y^{2}-P(x)\right| \leqslant 2|x| \text { if and only if }\left|x^{2}-P(y)\right| \leqslant 2|y|
\tag{1}
$$

Determine all possible values of $P(0)$.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1970
task_id: 2ca12d04-d771-5127-9235-1b94c111f613
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1970
task_revision_id: 1
config / split: OE_TO_maths_en_COMP / train

Official source

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