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2014 China Second Round Olympiad Second Part Problem 4

Source: 2014 China Second Round Olympiad

August 5, 2015
number theory

Problem Statement

Let x1,x2,…,x2014x_1,x_2,\dots,x_{2014} be integers among which no two are congurent modulo 20142014. Let y1,y2,…,y2014y_1,y_2,\dots,y_{2014} be integers among which no two are congurent modulo 20142014. Prove that one can rearrange y1,y2,…,y2014y_1,y_2,\dots,y_{2014} to z1,z2,…,z2014z_1,z_2,\dots,z_{2014}, so that among x1+z1,x2+z2,…,x2014+z2014x_1+z_1,x_2+z_2,\dots,x_{2014}+z_{2014} no two are congurent modulo 40284028.
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