CMO 1993 Problem 1

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 $3n$ .