Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers: $a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.
Problem
Source: China Mathematical Olympiad 1993 problem1
Tags: geometry, geometric transformation, number theory unsolved, number theory