MathDB

Consider all polynomials

P(x)

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

x

and

y

one has |y^2-P(x)|\le 2|x| \text{if and only if} |x^2-P(y)|\le 2|y|. Determine all possible values of

P(0)

.
Proposed by Belgium

Problem Statement