IN-TST 2009 Problem 9

Let $f(x)=\sum_{k=1}^n a_k x^k$ and $g(x)=\sum_{k=1}^n \frac{a_k x^k}{2^k -1}$ be two polynomials with real coefficients. Let g(x) have $0,2^{n+1}$ as two of its roots. Prove That $f(x)$ has a positive root less than $2^n$ .