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Belarusian Mathematical Olympiad

Algebra Difficulty 5.2 AIME, harder Prove it Belarus

Does there exist a function $f(x): \mathbb{R} \to \mathbb{R}$ satisfying the equality
$f(|x|) + |f(x)| = x$
for all real numbers $x$ ?

Solution

Answer: no, such function doesn't exist.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.