US-TST 2005 Problem 4

Consider the polynomials $$f(x) =\sum_{k=1}^{n}a_{k}x^{k} \text{and} g(x) =\sum_{k=1}^{n}\frac{a_{k}}{2^{k}-1}x^{k},$$ where $a_{1},a_{2},\ldots,a_{n}$ are real numbers and $n$ is a positive integer. Show that if 1 and $2^{n+1}$ are zeros of $g$ then $f$ has a positive zero less than $2^{n}$ .