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Suppose that a_1, a_2, \ldots, a_n are integers such that n\mid a_1 + a_2 + \ldots + a_n.
Prove that there exist two permutations \left(b_1,b_2,\ldots,b_n\right) and \left(c_1,c_2,\ldots,c_n\right) of \left(1,2,\ldots,n\right) such that for each integer i with 1\leq i\leq n, we have
n\mid a_i - b_i - c_i

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