Determine all polynomials $P(x)$ with real coefficients and which satisfy the following properties: i) $P(0) = 1$ ii) for any real numbers $x$ and $y,$ \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\]
Problem
Source: 2015 Brazil IMO TST 4.3 / easier version of IMO Shortlist 2014 A5
Tags: polynomial, algebra