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last four digits of Q(P(-1)) if P(Q(x)) = P(x)Q(x) and P(1) = P(-1) = 2019

Source: 2019 Dürer Math Competition Finals Day2 E+12 https://artofproblemsolving.com/community/c1621835_2019_

January 5, 2022

number theorypolynomiallast digitsDigits

Problem Statement

P

and

Q

are two different non-constant polynomials such that

P(Q(x)) = P(x)Q(x)

and

P(1) = P(-1) = 2019

. What are the last four digits of

Q(P(-1))