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Polynomials and primes

Source: 2020 Korean MO winter camp Test 1 P3

September 7, 2020
algebrapolynomial

Problem Statement

Find all integer coefficient polynomials QQ such that
[*] Q(n)≥1Q(n)\ge 1 ∀n∈Z+\forall n\in \mathbb{Z}_+. [*] Q(mn)Q(mn) and Q(m)Q(n)Q(m)Q(n) have the same number of prime divisors ∀m,n∈Z+\forall m,n\in\mathbb{Z}_+.
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