5

Part of 2020 Serbia National Math Olympiad

Problems(1)

Number theory functional equation

Source: 2020 Serbian MO, Problem 5

For a natural number

n

v_2(n)

we denote the largest integer

k\geq0

2^k|n

. Let us assume that the function

f\colon\mathbb{N}\to\mathbb{N}

meets the conditions:

(i)

f(x)\leq3x

for all natural numbers

x\in\mathbb{N}

(ii)

v_2(f(x)+f(y))=v_2(x+y)

for all natural numbers

x,y\in\mathbb{N}

.
Prove that for every natural number

a

there exists exactly one natural number

x

f(x)=3a

.

number theoryfunctional equationfunctionalgebra