2003 Putnam A6

For a set $S$ of nonnegative integers, let $r_S(n)$ denote the number of

ordered pairs $(s_1, s_2)$ such that $s_1 \in S$, $s_2 \in S$, $s_1 \ne

s_2$,

and $s_1 + s_2 = n$. Is it possible to partition the nonnegative

integers into two sets $A$ and $B$ in such a way that $r_A(n) = r_B(n)$

for all $n$ ?

对于一组非负整数 $S$，令 $r_S(n)$ 表示

有序对 $(s_1, s_2)$ 使得 $s_1 \in S$, $s_2 \in S$, $s_1 \ne

s_2$,

$s_1 + s_2 = n$。是否可以划分非负数

将整数分为两个集合 $A$ 和 $B$，使得 $r_A(n) = r_B(n)$

对于所有 $n$ ？