Disclaimer: The solutions we've shared are just one exciting approach, and there are surely many other wonderful methods out there. We’d love to hear your alternative solutions in the community thread below, so let's keep the creativity flowing!

For the sake of contradiction, assume that such a partition is possible.

From the conditions, if $a,~b$ are in a set and $ab$, $\frac{a}{b}$ are in the interval $[3,~3^5]$, they are in the opposite set.

Let the sets be $A,~B$ and assume that 3 is in $A$.

So, $3\cdot3$ is in $B$.

So, $3^2\cdot3^2=3^4$ is in $A$.

$A$ has $3^4,~3$. So, $3^4\cdot3=3^5$ and $\frac{3^4}{3}=3^3$ are in $B$.

We have $3^2,~3^3,~3^5$ in $B$. But, $3^2\cdot3^3=3^5$. Contradiction.

So, there is no such partition.