You can start by choosing an origin, and then define $r_1$, $r_2$, and $r_3$ in terms of of the $x$ and $y$ coordinate of this system. Then, realize that $\vec{B}$ has two components, which should both be zero for the magnetic field to vanish. You get two equations (one per component) in two unknowns (the coordinates), which you should be able to solve. You'll get two solution, one on either side of the horizontal axis.
A Note About A Certain Simplifiation
When we say that the field between these inductors is homogeneous, it's actually a simplification. These inductors have an effectively zero magnetic field outside of them because the field produced from one segment of wire effectively cancels out the field produced from a segment on the other side. This simplification lets us say: outside of an inductor, there is no magnetic field from that inductor.
This simplification breaks down as you get close to the solenoid/inductor. To be more specific, when the distance from one side of the inductor to the other diverges from zero, this simplification no longer holds. In practice, you rarely do anything where this simplification doesn't apply.
The Answer
Given the simplification that the magnetic field produced from a solenoid is zero when outside the coils of that solenoid, you can say that there is no magnetic field from the interior inductor in that space. Therefore it is as if that interior solenoid is not there. Assigning the variables to the larger, exterior solenoid with a subscript 1, you get $B = B_{1_{in}} = \frac{\mu\mu_0}l{I_1n_1}$ when $R_2<r<R_1$.
Best Answer
The field of a magnetic dipole is, at long distances (in some naturalized unit convention), equal to
$$ B_i = \partial_i ({\mu \cdot r \over 4\pi r^2})$$
where $\mu$ is the magnetic moment vector (the current times area of the loop, in the direction perpendicular to the area of the loop, times N for a solenoid with N windings, or just a given value for a fixed magnet). This form is universal for all magnetic dipoles at long distances, so it's the same for small loops, for small magnets and for small solenoids.
The form is easy to understand, because it's the form of the field for an electric dipole
The response of a second rigid magnetic dipole to this field is a torque which tends to align the magnetic moment along the direction of the field, plus a force proportional to the field gradient. These are also determined by the magnetic dipole of the magnet.
$$ T = \mu \times B $$
$$ F_j = - \mu^i \partial_i B_j $$
The first tends to align the magnetic dipole with the field, and this tends to happen quickly, the second moves the aligned dipole to the region of stronger field, and this happens more slowly.
These equations give a complete determination of the forces and torques acting between two dipoles, but if you substitute in the dipole field, the expressions become complicated and unilluminating. The picture is that there is a $1/r^3$ magnetic field which will align magnets so that their magnetic moment points along it (in the presence of dissipation), and then will lead to an attractive force as the aligned magnet drifts to a region of stronger field.