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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)| \leq 2 |x| \quad \text{if and only if} \quad |x^2 - P(y)| \leq 2 |y| \text{.}

Determine all possible values of P(0).

(Belgium)

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