We consider the problem of a fixed impurity coupled to a small number $N$ of noninteracting bosons. We focus on impurity-boson interactions that are mediated by a closed-channel molecule, as is the case for tuneable interatomic interactions in cold-atom experiments. We show that this two-channel model can be mapped to a boson model with effective boson-boson repulsion, which enables us to solve the three-body ($N=2$) problem analytically and determine the trimer energy for impurity-boson scattering lengths $a>0$. By analyzing the atom-dimer scattering amplitude, we find a critical scattering length $a^∗$ at which the atom-dimer scattering length diverges and the trimer merges into the dimer continuum. We furthermore calculate the tetramer energy exactly for $a>0$ and show that the tetramer also merges with the continuum at $a^∗$. Indeed, since the critical point $a^∗$ formally resembles the unitary point $1/a=0$, we find that all higher-body bound states (involving the impurity and $N>1$ bosons) emerge and disappear at both of these points. We show that the behavior at these “multibody resonances” is universal, since it occurs for any model with an effective three-body repulsion involving the impurity. Thus we see that the fixed-impurity problem is strongly affected by a three-body parameter even in the absence of the Efimov effect.