Let
![f: \mathbb{R}\to\mathbb{N}](/media/m/a/9/a/a9acd29c18e4afef1648846ce1a4773e.png)
be a function which satisfies
![f\left(x + \dfrac{1}{f(y)}\right) = f\left(y + \dfrac{1}{f(x)}\right)](/media/m/8/8/8/888186f030c8f9094bdd6504bfecf337.png)
for all
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
,
![y\in\mathbb{R}](/media/m/0/d/2/0d2137a3c3fee7b8818d1567a1c77ef4.png)
. Prove that there is a positive integer which is not a value of
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
.
Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithania
%V0
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