3

Part of 2015 Brazil Team Selection Test

Problems(1)

y^2-P(x)| <= 2|x| iff |x^2-P(y)|<=2|y|. P(0) = 1

Source: 2015 Brazil IMO TST 4.3 / easier version of IMO Shortlist 2014 A5

Determine all polynomials

P(x)

with real coefficients and which satisfy the following properties: i)

P(0) = 1

ii) for any real numbers

x

y,

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

polynomialalgebra