IMC 2025 · Problem 6

Let $f \colon (0, \infty) \to \mathbb{R}$ be a continuously differentiable function, and let $b > a > 0$ be real numbers such that $f(a) = f(b) = k$. Prove that there exists a point $\xi \in (a, b)$ such that

$$f(\xi) - \xi f'(\xi) = k.$$