2014 IMO Shortlist A5

Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has $$\left|y^{2}-P(x)\right| \leq 2|x| \quad \text { if and only if } \quad\left|x^{2}-P(y)\right| \leq 2|y|$$ Determine all possible values of $P(0)$.

考虑所有具有实数系数的多项式 $P(x)$ 具有以下属性：对于任意两个实数 $x$ 和 $y$，其中一个有 $$\left|y^{2}-P(x)\right| \leq 2|x| \quad \text { 当且仅当 } \quad\left|x^{2}-P(y)\right| \leq 2|y|$$ Determine all possible values of $P(0)$.