IMO 2020 Shortlist N4
For any odd prime p and any integer n, let dppnq P t0,1,...,p ´ 1u denote the remainder when n is divided by p. We say t...
Category: Number Theory
Problem
For any odd prime p and any integer n, let dppnq P t0,1,...,p ´ 1u denote the
remainder when n is divided by p. We say that pa0,a1,a2,...q is a p-sequence, if a0 is a positive
integer coprime to p, and an1 “ an dppanq for n ě 0.
(a) Do there exist infinitely many primes p for which there exist p-sequences pa0,a1,a2,...q and
pb0,b1,b2,...q such that an ą bn for infinitely many n, and bn ą an for infinitely many n?
(b) Do there exist infinitely many primes p for which there exist p-sequences pa0,a1,a2,...q and
pb0,b1,b2,...q such that a0 ă b0, but an ą bn for all n ě 1?
(United Kingdom)