We present exact results for magnetic impurities in nanoscopic systems with focusing properties. We analyze the spectroscopic and magnetic properties of Kondo, intermediate valence, and magnetic impurities on a sphere with a metallic surface. Exact calculations show the occurrence of spectroscopic and magnetic mirages at the antipodes of the impurity location. Comparison with calculations performed using effective models validates previous calculations of spectroscopic mirages. Our results predict the existence of a strong magnetic mirage in the experimentally realizable elliptic corral.