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Part of 2001 Czech And Slovak Olympiad IIIA

Problems(1)

P(x)^2 +P(-x) = P(x^2)+P(x) , polynomial

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 2001 p1

Determine all polynomials

P

such that for every real number

x

P(x)^2 +P(-x) = P(x^2)+P(x)

functional equationfunctionalpolynomialalgebra