ILL 1977 Problem 37

Let $A_1,A_2,\ldots ,A_{n+1}$ be positive integers such that $(A_i,A_{n+1})=1$ for every $i=1,2,\ldots ,n$ . Show that the equation $$x_1^{A_1}+x_2^{A_2}+\ldots + x_n^{A_n}=x_{n+1}^{A_{n+1} }$$ has an infinite set of solutions $(x_1,x_2,\ldots , x_{n+1})$ in positive integers.