For each subset $S$ of $\mathbb{N}$, with $r_S (n)$ we denote the number of ordered pairs $(a,b)$, $a,b\in S$, $a\neq b$, for which $a+b=n$. Prove that $\mathbb{N}$ can be partitioned into two subsets $A$ and $B$, so that $r_A(n)=r_B(n)$ for $\forall$ $n\in \mathbb{N}$.
Problem
Source: II International Festival of Young Mathematicians Sozopol 2011, Theme for 10-12 grade
Tags: combinatorics, set theory