IMO 2022 Shortlist N6
Let Q be a set of prime numbers, not necessarily finite. For a positive integer n consider its prime factorisation; defi...
Category: Number Theory
Problem
Let Q be a set of prime numbers, not necessarily finite. For a positive integer n
consider its prime factorisation; define ppnq to be the sum of all the exponents and qpnq to be
the sum of the exponents corresponding only to primes in Q. A positive integer n is called
special if ppnq ppn 1q and qpnq qpn 1q are both even integers. Prove that there is a
constant c ą 0 independent of the set Q such that for any positive integer N ą 100, the number
of special integers in r1,Ns is at least cN.
(For example, if Q “ t3,7u, then pp42q “ 3, qp42q “ 2, pp63q “ 3, qp63q “ 3, pp2022q “ 3,
qp2022q “ 1.)
(Costa Rica)