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Number theory functional equation

Source: 2020 Serbian MO, Problem 5

September 26, 2020

number theoryfunctional equationfunctionalgebra

Problem Statement

For a natural number

n

, with

v_2(n)

we denote the largest integer

k\geq0

such that

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

such that

f(x)=3a

.

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