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Philippines Mathematical Olympiad

Algebra Difficulty 5.5 AIME, harder Prove it Philippines

Problem:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function that satisfies the functional equation
$f(x-y)=2009 f(x) f(y)$
for all $x, y \in \mathbb{R}$ . If $f(x)$ is never zero, what is $f(\sqrt{2009})$ ?

Solution

Solution:
$\frac{1}{2009}$

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