N2

Part of 2023 Romanian Master of Mathematics Shortlist

Problems(1)

Integer polynomial and decimal digits

Source: RMM 2023 Shortlist N2

For every non-negative integer

k

S(k)

denote the sum of decimal digits of

k

P(x)

Q(x)

be polynomials with non-negative integer coecients such that

S(P(n)) = S(Q(n))

for all non-negative integers

n

. Prove that there exists an integer

t

P(x) - 10^tQ(x)

is a constant polynomial.

algebrapolynomialnumber theoryRMM Shortlistdecimal representation