IMC 2006 · Problem 6

Find all sequences $a_0, a_1, \dots, a_n$ of real numbers where $n \ge 1$ and $a_n \ne 0$, for which the following statement is true:

If $f : \mathbb{R} \to \mathbb{R}$ is an $n$ times differentiable function and $x_0 < x_1 < \dots < x_n$ are real numbers such that $f(x_0) = f(x_1) = \dots = f(x_n) = 0$ then there exists an $h \in (x_0, x_n)$ for which

$$a_0 f(h) + a_1 f'(h) + \dots + a_n f^{(n)}(h) = 0.$$