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Very beautiful and easy

Source: Romanian mo 1998

December 10, 2005
algebra proposedalgebra

Problem Statement

Suppse that n≥2n\geq 2 and 0<x1<x2<...<xn0<x_1<x_2<...<x_n are integer numbers. We denote that :Sk=∑A⊂{x1,x2,...,xn}1∏a∈Aa,k=1,2,...,n. S_k=\sum_{A\subset \{x_1,x_2,...,x_n\}} \frac{1}{\prod_{a\in A}a} , k=1,2,...,n. (where AA is a non-empty subset). Show that if Sn,Sn−1S_n ,S_{n-1} were positive integer numbers , then ∀k:Sk\forall k : S_k is a positive integer.
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