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> If $f\in C^1(\mathbb R)$ satisfies $f'(x)>f(f(x))$ for all $x\in\mathbb R$, then $f(f(f(x)))\leq0$ for all $x\geq0$.
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I have some trouble to prove this. I wonder if there's some relations between this problem and the ODE $ f'(x)=f(f(x)) $. Could anybody provide a solution or some hints on this problem?



$f'(x)>f(f(x))$ implies $f(f(f(x)))\leq0$ for nonnegative $x$
Well, this is Problem 4 of Day 1 of IMC 2012, proposed by Tomáš Bárta from Prague, see [here](https://www.imc-math.org.uk/imc2012/IMC2012-day1-solutions.pdf)


