Test candidates' ability to construct original Lyapunov functions to prove stability (Lyapunov, asymptotic, exponential) of equilibria in nonlinear dynamical systems. Scope includes selection and synthesis of energy-like or composite functions, verification of positive-definiteness and radial unboundedness, computing Lyapunov derivatives, applying LaSalle and converse theorems, using Krasovskii or sum-of-squares approaches, and producing rigorous proofs for local and global stability for given examples and parametric families.