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 IMO ShortlistalgebraFunctional inequalityPolynomials