11.4. Find all natural numbers $n$ that can be represented as $n=\frac{x+\frac{1}{x}}{y+\frac{1}{y}}$, for some natural numbers $x$ and $y$.
Answer. $n=1$

Solution. Transform the equality in the condition to the form $n=\frac{\left(x^{2}+1\right) y}{\left(y^{2}+1\right) x}$. Note that the numbers $x^{2}+1$ and $x$, as well as the numbers $y^{2}+1$ and $y$ are coprime, so $x$ in the denominator can only cancel out with $y$ in the numerator, meaning $y$ is divisible by $x$ and, in particular, $y$ is not less than $x$. Similarly, $y^{2}+1$ in the denominator can only cancel out with $x^{2}+1$ in the numerator, so $y^{2}+1$ divides $x^{2}+1$, in particular, does not exceed it, from which it follows that $y$ does not exceed $x$. Therefore, $y$ equals $x$ and $n=1$. The number $n=1$ is obtained for any pair of equal $x$ and $y$, for example, when they are both equal to 1.

Grading criteria. A statement in the solution like “the numbers $x^{2}+1$ and $x$, as well as the numbers $x^{2}+1$ and $y$ are coprime” can be considered correct without detailed explanations. If there are no references to coprimality: the score will not exceed 2 points. If it is not explicitly stated that the number $n=1$ is still obtained for any pair of equal $x$ and $y$, for example, when they are equal to something specific: 1 point will be deducted.