International Mathematics Competition
for University Students
2018

IMC2018: Day 1, Problem 4

Problem 4. Find all differentiable functions \(\displaystyle f:(0,\infty)\to\RR\) such that

\(\displaystyle f(b)-f(a)=(b-a)f'\left(\sqrt{ab}\right) \quad \text{for all} \quad a,b>0. \tag2 \)

(Proposed by Orif Ibrogimov, National University of Uzbekistan)

Hint: Differentiate the equation and transform it to a differential equation.

  Solution