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Function preserves sum-free-ness

Source: Romanian Masters of Mathematics 2020, Problem 4

March 1, 2020

functionalgebraRMMRMM 2020

Problem Statement

Let

\mathbb N

be the set of all positive integers. A subset

A

of

\mathbb N

is sum-free if, whenever

x

and

y

are (not necessarily distinct) members of

A

, their sum

x+y

does not belong to

A

. Determine all surjective functions

f:\mathbb N\to\mathbb N

such that, for each sum-free subset

A

of

\mathbb N

, the image

\{f(a):a\in A\}

is also sum-free.
Note: a function

f:\mathbb N\to\mathbb N

is surjective if, for every positive integer

n

, there exists a positive integer

m

such that

f(m)=n

.

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