International Mathematics Competition
for University Students
2025

IMC2025: Day 1, Problem 2

Problem 2. Let \(\displaystyle f\colon\mathbb{R} \to \mathbb{R}\) be a twice continuously differentiable function, and suppose that \(\displaystyle \int_{-1}^{1}f(x)\,\mathrm{d}x=0\) and \(\displaystyle f(1)=f(-1)=1\). Prove that

\(\displaystyle \int_{-1}^{1} \left(f''(x)\right)^2\,\mathrm{d}x\geq 15 , \)

and find all such functions for which equality holds.

Alberto Cagnetta, Università degli Studi di Udine, Italy

  Solution