MathDB

Integer polynomial and decimal digits

Source: RMM 2023 Shortlist N2

February 29, 2024

algebrapolynomialnumber theoryRMM Shortlistdecimal representation

Problem Statement

For every non-negative integer

k

let

S(k)

denote the sum of decimal digits of

k

. Let

P(x)

and

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

such that

P(x) - 10^tQ(x)

is a constant polynomial.