Three problems for the interaction between N bodies to which exact solutions were obtained are discussed. In the first problem, the bodies are located axisymmetrically in a plane. Possible trajectories of the bodies here are ellipses, parabolas, hyperbolas, or straight lines. Results of solving this problem are further used for solving problems about Earth’s rotation and Mercury’s perihelion. In the second problem, the body system consists of several layers, and it rotates as an entity. In the third problem, the bodies are uniformly distributed over a sphere, and they move experiencing no mutual collisions. The latter problem allows the formation of several planets, for instance, one hundred planets resembling the Earth and moving under identical conditions with respect to the Sun. The latter possibility opens a way toward unrestricted mankind progress.