MathDB

3

Part of 2021 Philippine MO

Problems(1)

pmo problem 3

Source: PMO

3/19/2021
Denote by Q+\mathbb{Q}^+ the set of positive rational numbers. A function f:Q+→Qf : \mathbb{Q}^+ \to \mathbb{Q} satisfies
• f(p)=1f(p) = 1 for all primes pp, and
• f(ab)=af(b)+bf(a)f(ab) = af(b) + bf(a) for all a,b∈Q+ a,b \in \mathbb{Q}^+ .
For which positive integers nn does the equation nf(c)=cnf(c) = c have at least one solution cc in Q+\mathbb{Q}^+?
functional equationnumber theoryPMO