MathDB

P-Dop P-Dop

Source: ISL 2020 N4

July 20, 2021

number theoryIMO ShortlistIMO Shortlist 2020Sequence

Problem Statement

For any odd prime

p

and any integer

n,

let

d_p (n) \in \{ 0,1, \dots, p-1 \}

denote the remainder when

n

is divided by

p.

We say that

(a_0, a_1, a_2, \dots)

is a p-sequence, if

a_0

is a positive integer coprime to

p,

and

a_{n+1} =a_n + d_p (a_n)

for

n \geqslant 0.

(a) Do there exist infinitely many primes

p

for which there exist

p

-sequences

(a_0, a_1, a_2, \dots)

and

(b_0, b_1, b_2, \dots)

such that

a_n >b_n

for infinitely many

n,

and

b_n > a_n

for infinitely many

n?

(b) Do there exist infinitely many primes

p

for which there exist

p

-sequences

(a_0, a_1, a_2, \dots)

and

(b_0, b_1, b_2, \dots)

such that

a_0 <b_0,

but

a_n >b_n

for all

n \geqslant 1?

[I]United Kingdom

Back to Problems View on AoPS