The problem of constructing optimum or close to optimum nose-shapes of bodies of revolution of fixed aspect ratio in a supersonic flow is solved within the framework of a perfect (inviscid and non-heat-conducting) gas. Their contour includes the front face, that is, the boundary extremum section with respect to the length and, adjacent to it, the smooth, slightly sloping section that makes a corner. In case of low aspect ratios, the slightly sloping section is the result of the exact solution of a variational problem. In the case of aspect ratios which exceed a certain value, depending on the free-stream Mach number M ∞, the exact solution requires the introduction of small internal breaks with corner points where even the dominant one of these only has a weak effect on the drag value. Contours which are referred to as “close to optimum” do not satisfy the optimality condition, which defines the dominant corner. In the examples (1.2 ≤ M ∞ ≤ 10) for which calculations were carried out, conical nose shapes were found to be far worse that the optimum ones. For contours which are optimum in the approximation of Newton's formula and also, optimum blunt and pointed, power-law nose shapes, the situation occurs for low-aspect ratios and low supersonic Mach numbers (pointed, power-law contours can only be successfully constructed for fairly high aspect ratios). The fact that the front face is a section of a boundary extremum is shown by comparing the drags of bodies obtained with different permissible variations of the front face. An alternative proof, which is not limited by the actual form of the variation in the front face, can be obtained from the solution of the conjugate problem, formulated within the framework of the general method of Lagrange multipliers. This problem is also of interest in its own right, in particular, on account of the singularities, revealed during its formulation, in the reflection of the discontinuities of the Lagrange multipliers from the sonic line with parts of them becoming infinite at the point of reflect.