MathNet 2020 — Math Olympiad Vault — Part 05 Question 115

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 again sum-free.
India, Sutanay Bhattacharya