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China Mathematical Olympiad 1993 problem1

Source: China Mathematical Olympiad 1993 problem1

September 22, 2013
geometrygeometric transformationnumber theory unsolvednumber theory

Problem Statement

Given an odd nn, prove that there exist 2n2n integers a1,a2,⋯ ,ana_1,a_2,\cdots ,a_n; b1,b2,⋯ ,bnb_1,b_2,\cdots ,b_n, such that for any integer kk (0<k<n0<k<n), the following 3n3n integers: ai+ai+1,ai+bi,bi+bi+ka_i+a_{i+1}, a_i+b_i, b_i+b_{i+k} (i=1,2,⋯ ,n;an+1=a1,bn+j=bj,0<j<ni=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n) are of different remainders on division by 3n3n.
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