International Mathematics Competition
for University Students
2023

IMC2023: Day 2, Problem 7

Problem 7. Let \(\displaystyle V\) be the set of all continuous functions \(\displaystyle f\colon[0,1] \to \mathbb{R}\), differentiable on \(\displaystyle (0,1)\), with the property that \(\displaystyle f(0)=0\) and \(\displaystyle f(1)=1\). Determine all \(\displaystyle \alpha\in\mathbb{R}\) such that for every \(\displaystyle f\in V\), there exists some \(\displaystyle \xi\in(0,1)\) such that

\(\displaystyle f(\xi) + \alpha = f'(\xi). \)

Mike Daas, Leiden University

Hint: Find a function \(\displaystyle h\in V\) such that \(\displaystyle h'-h\) is constant, then apply Rolle's theorem to \(\displaystyle f-h\).

Alternatively, you can apply Cauchys's mean value theorem with some auxiliary functions.

  Solution