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Let n be an odd integer greater than 1 and let c_1, c_2, \ldots, c_n be integers. For each permutation a = (a_1, a_2, \ldots, a_n) of \{1,2,\ldots,n\}, define S(a) = \sum_{i=1}^n c_i a_i. Prove that there exist permutations a \neq b of \{1,2,\ldots,n\} such that n! is a divisor of S(a)-S(b).

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