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Belarusian Mathematical Olympiad

Algebra Difficulty 4.9 AIME Prove it Belarus

Find all functions $f(x): \mathbb{R} \to \mathbb{R}$ satisfying the equality
$x = -\frac{1}{2}f(|x|) + |f(x)|$
for all real numbers $x$ .

Solution

$f(x) = 2x$ for all $x \ge 0$ and $f(x) = 0$ for all $x < 0$ .

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