Difference between revisions of "2011 IMO Problems/Problem 3"

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Let f : R R be a real-valued function defined on the set of real numbers that satisfies f(x + y) yf(x) + f(f(x)) for all real numbers x and y. Prove that f(x) = 0 for all x≤ 0.
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Let <math>f: \mathbb R \to \mathbb R</math> be a real-valued function defined on the set of real numbers that satisfies <cmath> f(x + y) \le yf(x) + f(f(x)) </cmath> for all real numbers <math>x</math> and <math>y</math>. Prove that <math>f(x) = 0</math> for all <math>x \le 0</math>.

Revision as of 13:38, 27 November 2011

Let $f: \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies \[f(x + y) \le yf(x) + f(f(x))\] for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \le 0$.

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