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Integral functional equation on reals

2175ExpertCalculusFunctions

Romanian Mathematical Olympiad

Determine all functions $f: \mathbb{R} \to \mathbb{R}$, integrable over any bounded interval, satisfying the condition $$\int_{x-y}^{x+y} f(t) \, dt = y(f(x+y) + f(x-y)), \quad \text{for all real numbers } x \text{ and } y.$$
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