Question 55
Let $f(x) = \sum_{k=1}^{n} a_k x^k$ and $g(x) = \sum_{k=1}^{n} \frac{a_k}{2^k - 1} x^k$ be two polynomials with real coefficients, where $n \ge 3$. Suppose $1, 2^{n+1}$ are roots of $g(x) = 0$. Prove that $f(x) = 0$ has a positive root smaller than $2^n$.