  We study a strongly interacting system of N identical bosons and one impurity
in a one-dimensional trap. First, we assume that the particles have identical
masses and analyze the corresponding set-up. After that, we study the influence
of a small mass asymmetry on our analysis. In particular, we discuss how the
structure of the wave function and the degeneracy in the impenetrable regime
depend on the mass ratio and the shape of the trapping potential. To illustrate
our findings, we consider a four-body system in a box and in an oscillator. We
show that in the former case the system has the smallest energy when a heavy
(light) impurity is close to the edge (center) of the trap. And we demonstrate
that the opposite is true in the latter case.
