Problem

Source: 2016 Ukraine TST

Tags: algebra, polynomial, calculus, TST



Let $a_1,\ldots, a_n$ be real numbers. Define polynomials $f,g$ by $$f(x)=\sum_{k=1}^n a_kx^k,\ g(x)=\sum_{k=1}^n \frac{a_k}{2^k-1}x^k.$$Assume that $g(2016)=0$. Prove that $f(x)$ has a root in $(0;2016)$.