We take a real analysis approach to characterize the symmetric mixed-strategy equilibrium (SMSE) of rent-seeking contests for any intermediate value of the decisiveness parameter, allowing for both the ratio-form and the difference-form of contest success functions (CSF) that are axiomatized in Skaperdas (1996). For the ratio-form Tullock contest, we reproduce the results obtained by Ewerhart (2015) by invoking sophisticated complex analysis that SMSE is supported by a countably infinite set of pure strategies that has zero bid as an accumulation point and that the rent is completely dissipated. For the Hirshleifer-style difference-form of CSF, we show that the support of SMSE is a finite set of pure strategies including zero bid as one mass point and that the rent is partially dissipated. The similarity and differences in the profile of SMSE between the two canonical forms of CSF in the literature of contests are thereby illuminated.