International Mathematics Competition
for University Students
2025

IMC2025: Day 2, Problem 6

Problem 6. Let \(\displaystyle f\colon (0,\infty) \to \mathbb{R}\) be a continuously differentiable function, and let \(\displaystyle b>a>0\) be real numbers such that \(\displaystyle f(a)=f(b) = k\). Prove that there exists a point \(\displaystyle \xi\in(a,b)\) such that

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

Alberto Cagnetta, Università degli Studi di Udine

  Solution