IMC 1994 · Problem 3

Let $f$ be a real-valued function with $n+1$ derivatives at each point of $\mathbb{R}$. Show that for each pair of real numbers $a$, $b$, $a < b$, such that

$$\ln\left( \frac{f(b) + f'(b) + \dots + f^{(n)}(b)}{f(a) + f'(a) + \dots + f^{(n)}(a)} \right) = b - a$$

there is a number $c$ in the open interval $(a,b)$ for which

$$f^{(n+1)}(c) = f(c).$$

Note that $\ln$ denotes the natural logarithm.