Within the limits of the ideal incompressible flow and Chaplygin gas models of the subsonic adiabatic motion of a perfect gas, exact solutions are constructed for the fundamental inverse variational boundary-value problem of aerohydrodynamics, namely, the problem of designing an airfoil of maximum life, on the assumption that the maximum velocity on its contour is limited. The term "variational inverse boundary-value problem" is used to designate a class of two-dimensional boundary-value problems with unknown boundaries, in which it is required to find both the solution of a partial differential equation and its domain of definition, where the latter satisfies some extremal property, and one boundary condition is specified on its boundary. The extremal property of the domain is expressed as the requirement that a certain functional be maximized or minimized (usually with further constraints). The existence and uniqueness of solutions is analysed, admissible domains of the parameters are indicated, examples are given of exact solutions, and an analysis is presented of the tendencies of the aerodynamic shapes being optimized to change when the theoretical angle of attack and maximum value of the velocity on the airfoil contour are varied. The so-called "shelf" distributions of velocity (with sections of constant velocity) are obtained as extremal.