MathDB

12

Part of 2019 Durer Math Competition Finals

Problems(2)

arrange the numbers 1, 2, 3, .. , 15 in some order

Source: 2019 Dürer Math Competition Finals Day2 E12 https://artofproblemsolving.com/community/c1621835_2019_

1/5/2022
How many ways are there to arrange the numbers 1,2,3,..,151, 2, 3, .. , 15 in some order such that for any two numbers which are 22 or 33 positions apart, the one on the left is greater?
combinatorics
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_

1/5/2022
PP and QQ are two different non-constant polynomials such that P(Q(x))=P(x)Q(x)P(Q(x)) = P(x)Q(x) and P(1)=P(−1)=2019P(1) = P(-1) = 2019. What are the last four digits of Q(P(−1))Q(P(-1))?
number theorypolynomiallast digitsDigits