IMC 2007 · Problem 6

Let $f \ne 0$ be a polynomial with real coefficients. Define the sequence $f_0, f_1, f_2, \dots$ of polynomials by $f_0 = f$ and $f_{n+1} = f_n + f_n'$ for every $n \ge 0$. Prove that there exists a number $N$ such that for every $n \ge N$, all roots of $f_n$ are real.