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2001 IMO Shortlist Problems/C2

Revision as of 18:19, 20 August 2008 by Minsoens (talk | contribs) (New page: == Problem == Let <math>n</math> be an odd integer greater than 1 and let <math>c_1, c_2, \ldots, c_n</math> be integers. For each permutation <math>a = (a_1, a_2, \ldots, a_n)</math> of <...)
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Problem

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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