For any odd prime p and any integer n, let dp(n)∈{0,1,…,p−1} denote the remainder when n is divided by p. We say that (a0,a1,a2,…) is a p-sequence, if a0 is a positive integer coprime to p, and an+1=an+dp(an) for n⩾0.
(a) Do there exist infinitely many primes p for which there exist p-sequences (a0,a1,a2,…) and (b0,b1,b2,…) such that an>bn for infinitely many n, and bn>an for infinitely many n?
(b) Do there exist infinitely many primes p for which there exist p-sequences (a0,a1,a2,…) and (b0,b1,b2,…) such that a0<b0, but an>bn for all n⩾1?[I]United Kingdom number theoryIMO ShortlistIMO Shortlist 2020Sequence