This paper presents investigation of one class of variational problems of minimization of the drag and heat fluxes to the surfaces of three dimensional bodies. In a class of slender bodies possessing homothetic property the initial optimization problem may be reduced to the problem of finding optimal transverse contours. The investigation of the problem of determining the optimal transversal contour has shown that there exist the solutions composed of n identical cycles. A distinguishing feature of the suggested approach is that the minimization procedure includes not only a search for each extremal segment but also the number of these segments. This leads to the additional optimizing condition on the number of cycles. Joint integration of the Euler-Lagrange equation and the condition mentioned above has permitted the author to obtain three classes of analytical solutions.