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Source: Romanian IMO Team Selection Test TST 2003, problem 3

September 24, 2005

Problem Statement

Let n,kn,k be positive integers such that nk>(k+1)!n^k>(k+1)! and consider the set M=\{(x_1,x_2,\ldots,x_n)\dvd x_i\in\{1,2,\ldots,n\},\ i=\overline{1,k}\}. Prove that if A⊂MA\subset M has (k+1)!+1(k+1)!+1 elements, then there are two elements {α,β}⊂A\{\alpha,\beta\}\subset A, α=(α1,α2,…,αn)\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n), β=(β1,β2,…,βn)\beta=(\beta_1,\beta_2,\ldots,\beta_n) such that (k+1)!∣(β1−α1)(β2−α2)⋯(βk−αk). (k+1)! \left| (\beta_1-\alpha_1)(\beta_2-\alpha_2)\cdots (\beta_k-\alpha_k) \right. .
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