Which form is the correct when asking someone how a certain issue was solved:
How did you manage the door problem?
or
How did you managed the door problem?
I'd go with "managed", but a native speaker told me "manage" was the correct form, although she couldn't explain why.
I searched my memory and the internet for what this structure is called, but I could not find it.
Did you [unconjugated verb] is how English is. Unconjugated means the verb, without any -ing, -ed, etc. For instance, you wouldn't say Did you ate my food?, but rather, Did you eat my food?. Typically, if not exclusively, this mechanic is used to turn a statement into a question:
You ate my food. ---> Did you eat my food?
I suppose the word did, being the past tense of do, is taking care of the tense already, so we use the most simple version of the verb, avoiding redundancy.
See this question from the ELL Stack Exchange. You may find that Stack Exchange more useful than this one, anyway.
"Do" is what we called "the dummy operator" along with did, does, etc. It's a part of auxiliaries.
When we introduce a sentence but we don't have any auxiliary for it we would then use dummy operator to do the task.
For example:
Jack cleans the bathroom.
Notice that the sentence does not have any auxiliary so If you are the manager and want to make sure Jack is doing his job from his co-worker, you have to come up with a question, therefore, you must apply the dummy operator which is:
Does Jack clean the bathroom today? (Not assuming he finished)
Or
did Jack clean the bathroom? ( assuming he finished)
Now, there are four-based forms of this dummy operator:
I did phone. Did you phone? I did not phone. Samantha phoned, I did too.
Answering your question: how did you manage the floor problem.
Because it's a question type sentence "how" ,so we put the auxiliary in front of the the subject. "Did you"
And because the event is passed and so you refer to the past which is "did"
If it is just a sentence without question would be : you did manage the floor problem.
You're just converting the sentence into a question type sentence. Also you asking the past which is why "did" applied.
Hope this helps.
