Problem

Source: Romanian mo 1998

Tags: algebra proposed, algebra



Suppse that $n\geq 2$ and $0<x_1<x_2<...<x_n$ are integer numbers. We denote that :\[ S_k=\sum_{A\subset \{x_1,x_2,...,x_n\}} \frac{1}{\prod_{a\in A}a} , k=1,2,...,n. \] (where $A$ is a non-empty subset). Show that if $S_n ,S_{n-1}$ were positive integer numbers , then $\forall k : S_k$ is a positive integer.