International Mathematics Competition for University Students

July 31 – August 6 2017, Blagoevgrad, Bulgaria

Home

Ivan is Watching You

Day 1
    Problem 1
    Problem 2
    Problem 3
    Problem 4
    Problem 5

Day 2
    Problem 6
    Problem 7
    Problem 8
    Problem 9
    Problem 10

Results
    Individuals
    Teams
    Leaders

Download
    Day 1 questions
    Day 1 solutions
    Day 2 questions
    Day 2 solutions
    Closing Ceremony
        Presentation

Official IMC site

Problem 2

2. Let $f\colon\mathbb{R}\to(0,\infty)$ be a differentiable function, and suppose that there exists a constant $L>0$ such that $$ \bigl|f'(x)-f'(y)\bigr| \leq L\bigl|x-y\bigr| $$ for all $x,y$. Prove that $$ \big(f'(x)\big)^2 < 2Lf(x) $$ holds for all $x$.

Proposed by: Jan Ĺ ustek, University of Ostrava

Hint: Integrate $f'$ over an interval $[x,x+\Delta]$.

  Solution