The correct sentence is “Did you manage to finish?” and not “Did you managed to finish?”.
The reason is that “manage” is a regular verb, and in the simple past tense, it should be conjugated as “managed.” In English, verbs typically follow a consistent pattern in past tense. For example, verbs like “work” become “worked,” and “manage” follows the same rule by becoming “managed.” Additionally, when forming questions in the past tense, we use the simple past form of the verb without adding an extra “d” or any other suffix, making “manage” the correct form.