MathDB

6

Part of 2018 India National Olympiad

Problems(1)

INMO 2018 -- Problem #6

Source: INMO 2018

1/21/2018
Let N\mathbb N denote set of all natural numbers and let f:N→Nf:\mathbb{N}\to\mathbb{N} be a function such that
(a)f(mn)=f(m).f(n)\text{(a)} f(mn)=f(m).f(n) for all m,n∈Nm,n \in\mathbb{N};
(b)m+n\text{(b)} m+n divides f(m)+f(n)f(m)+f(n) for all m,n∈Nm,n\in \mathbb N.
Prove that, there exists an odd natural number kk such that f(n)=nkf(n)= n^k for all nn in N\mathbb{N}.
number theoryfunctional equation