International Mathematics Competition for University Students

July 31 – August 6 2017, Blagoevgrad, Bulgaria

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

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

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

Official IMC site

Problem 9

9. Define the sequence $f_1,f_2,\ldots:[0,1)\to \RR$ of continuously differentiable functions by the following recurrence: $$ f_1=1; \qquad \quad f_{n+1}'=f_nf_{n+1} \quad\text{on $(0,1)$}, \quad \text{and}\quad f_{n+1}(0)=1. $$

Show that $\lim\limits_{n\to \infty}f_n(x)$ exists for every $x\in [0,1)$ and determine the limit function.

Proposed by: Tomáš Bárta, Charles University, Prague

  Hint    Solution