2011 IMO Problems/Problem 3: Difference between revisions
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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>. | 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>. | ||
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Revision as of 23:05, 10 October 2013
Let 
be a real-valued function defined on the set of real numbers that satisfies ![\[f(x + y) \le yf(x) + f(f(x))\]](http://latex.artofproblemsolving.com/6/3/e/63e20e9b2787ab8d177c6406d9db72cdbada44ad.png)
for all real numbers 
and 
. Prove that 
for all 
.
Solution
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