This paper presents an application of the adjoint equation method, a well-established method for aerodynamic shape optimization in external flow, to the internal-flow problem of engine inlet design. A serpentine shaped duct with prescribed entrance and exit geometries is considered, where the design objective is to optimize the curvature and cross-sections of the duct in order to reduce flow distortion at the exit. In an indirect formulation of the problem, the wall pressure drag is chosen as objective function, under which the adjoint method can be formulated. Inviscid flow is considered, with the Euler equation and its adjoint as governing equations. The results show that this leads in return to an improved uniformity of the exit-face total pressure, albeit with slight loss in the average total pressure.