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2010 smo (2)

Source: 2010 China South East Mathematical Olympiad

July 18, 2011
algebrapolynomialmodular arithmeticLaTeXnumber theorynumber theory unsolved

Problem Statement

For any set A={a1,a2,⋯ ,am}A=\{a_1,a_2,\cdots,a_m\}, let P(A)=a1a2⋯amP(A)=a_1a_2\cdots a_m. Let n=(201099)n={2010\choose99}, and let A1,A2,⋯ ,AnA_1, A_2,\cdots,A_n be all 9999-element subsets of {1,2,⋯ ,2010}\{1,2,\cdots,2010\}. Prove that 2010∣∑i=1nP(Ai)2010|\sum^{n}_{i=1}P(A_i).
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