Problem

Source: 2023 China South-east MO Grade 10 P2

Tags: number theory



$A$ is a non-empty subset of positive integers. Let $$f(A)=\{abc-b-c+2\vert a,b,c\in A\}$$Determine all integers $n$ greater than $1$ so that we can divide the set of positive integers into $A_1, A_2, \dots, A_n$ ($A_i\neq \emptyset (i=1, 2, \dots , n)$, $\forall 1\le i < j \le n, A_i\cap A_j = \emptyset$ and $\bigcup_{i=1}^{n} A_i=\mathbb{N}^*$) satisfy that $\forall 1\le i\le n, f(A_i) \subseteq A_i$.