The paper addresses the problem of packing pairwise unequal circles into the circle of minimal radius centered at the origin. This problem with an additional balance condition is considered as well. The corresponding optimization models are highly nonconvex. Symmetry breaking constraints are developed to reduce the number of solutions. It is proved that these additional linear constraints avoid solutions coming from rotations and certain reflections. In addition, problems with subsets of equally sized circles are dealt with as well. Computational experiments based on small benchmark instances demonstrate significant savings in runtime for the global solver BARON if models with symmetry breaking constraints are employed.