International Mathematics Competition
for University Students
2019

IMC2019: Day 1, Problem 3

Problem 3. Let \(\displaystyle f:(-1,1)\to\RR\) be a twice differentiable function such that

\(\displaystyle {2f'(x)+xf''(x)\geq1} \quad\text{for \(\displaystyle x\in(-1,1)\)}. \)

Prove that

\(\displaystyle \int_{-1}^1xf(x)\,\mathrm{d}x\geq\frac{1}{3}. \)

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and

Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan

Hint: \(\displaystyle 2f'(x)+xf''(x)\) is the second derivative of a certain function.

  Solution