BG-MO 1984 Problem 5

Let $0<x_i<1$ and $x_i+y_i=1$ for $i=1,2,\ldots,n$ . Prove that $$(1-x_1x_2\cdots x_n)^m+(1-y_1^m)(1-y_2^m)\cdots(1-y_n^m)>1$$ for any natural numbers $m$ and $n$ .