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polynomial FE, prove existence in p(x)p(4-x)=p(x(4-x))

Source: Mongolia MO 2001 Teachers P1

April 12, 2021

functional equationPolynomials

Problem Statement

Prove that for every positive integer

n

there exists a polynomial

p(x)

of degree

n

with real coefficients, having

n

distinct real roots and satisfying

p(x)p(4-x)=p(x(4-x))