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Let f: \mathbb{R}\to\mathbb{N} be a function which satisfies f\left(x + \dfrac{1}{f(y)}\right) = f\left(y + \dfrac{1}{f(x)}\right) for all x, y\in\mathbb{R}. Prove that there is a positive integer which is not a value of f.

Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithania

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