6. Find all pairs of non-zero (not necessarily positive) rational numbers $x, y$ that have the following property: any positive rational number can be represented as $\{r x\} / \{r y\}$ with a positive rational $r$.

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Answer: all pairs where $x y<0$ are suitable. Divide the plane into unit squares by lines of the integer lattice. Draw a line $\ell$ through the origin $O$ and the point $(x, y)$. Since $x, y \in \mathbb{Q}$, it has a rational slope and passes through some lattice node. A point of the form $(r x, r y)$ is any rational point on this line. Draw a line through such a point and the lower-left corner of the cell into which this point falls. The slope of this line is precisely $\{r x\} / \{r y\}$.

Align all unit squares that contain points of $\ell$. In the resulting square, $\ell$ will be represented by a finite number of segments parallel to the original line.

Suppose $x$ and $y$ are of the same sign. Then the resulting segments have a positive slope. One of them extends from the lower-left node of the square. It is clear that from this node, one can draw a ray with a positive rational slope $\lambda$ that intersects the upper or lower side of the square without encountering any of our segments. This number $\lambda$ cannot be represented in the required form.

On the contrary, let the numbers $x$ and $y$ be of opposite signs. Then our segments have a negative slope. One of them extends from the upper-left node of the square. It is not difficult to understand that one of these segments connects a point on the left side of the square with a point on its lower side. On the rational points of this segment, all possible positive rational slopes are realized!