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(a_1+a_2+...+a_n)!! / a_1!!a_2!!... a_n!! is an integer

Source: Mathcenter Contest / Oly - Thai Forum 2009 R2 p1 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 9, 2022
double factorialfactorialnumber theory

Problem Statement

For any natural nn , define n!!=(n!)!n!!=(n!)! e.g. 3!!=(3!)!=6!=7203!!=(3!)!=6!=720. Let a1,a2,...,ana_1,a_2,...,a_n be a positive integer Prove that (a1+a2+⋯+an)!!a1!!a2!!⋯an!!\frac{(a_1+a_2+\cdots+a_n)!!}{a_1!!a_2!!\cdots a_n!!} is an integer.
(nooonuii)
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