Vi bliver ved med at tale om problemet, indtil vi bliver enige.

Bliver ved med at is a fixed expression meaning keep doing something or continue to do something.

So:

• Vi bliver ved med at tale = We keep talking
• literally, it is something like we become/stay on with to talk, but you should learn it as one chunk.

A very useful pattern is:

• blive ved med at + infinitive

Examples:

• Han bliver ved med at spørge. = He keeps asking.
• De bliver ved med at grine. = They keep laughing.

No, they are part of two different expressions.

1. bliver ved med at tale
   Here bliver is part of the fixed phrase blive ved med at, meaning keep/continue to.

2. bliver enige
   Here bliver has its more basic meaning become.
   blive enige means become agreed or more naturally reach agreement / agree.

So even though the same word appears twice, it is doing different jobs in each part of the sentence.

Because after blive ved med Danish normally uses:

• at + infinitive

So:

• blive ved med at tale
• blive ved med at arbejde
• blive ved med at prøve

You cannot normally leave out the at here.

Tale om means talk about.

The verb tale often needs the preposition om when you mention the topic:

• tale om vejret = talk about the weather
• tale om problemet = talk about the problem

So om works like English about here.

Because problemet is the definite form: the problem.

In Danish:

• et problem = a problem
• problemet = the problem

The ending -et marks definiteness for many neuter nouns.

So:

• om problemet = about the problem

Indtil means until.

It introduces the point where the first action stops or changes:

• Vi bliver ved med at tale om problemet, indtil vi bliver enige.
• We keep talking about the problem until we agree.

So the talking continues up to the moment of agreement.

Because indtil introduces a subordinate clause.

In Danish subordinate clauses, the subject usually comes before the verb:

• indtil vi bliver enige
• fordi han kommer
• når de er færdige

So this is normal subordinate-clause word order.

Compare:

• main clause: Vi bliver enige
• subordinate clause: ... indtil vi bliver enige

In this sentence, the word order happens to look the same as in English.

Because enige is the plural form of the adjective enig.

Here the subject is vi = we, so the adjective must match that plural subject:

• jeg er enig = I agree
• han er enig = he agrees / is in agreement
• vi er enige = we agree / are in agreement

And with blive:

• vi bliver enige = we come to an agreement

So enige is used because vi is plural.

Very often, yes.

Danish commonly uses være enig and blive enig:

• være enig = be in agreement / agree
• blive enig = come to an agreement / end up agreeing

Examples:

• Jeg er enig. = I agree.
• Vi blev enige om planen. = We agreed on the plan.

In your sentence, bliver enige suggests reaching agreement after discussion.

Yes, often it could.

• tale = speak / talk
• snakke = talk / chat

In many everyday contexts they are very close.
So Vi bliver ved med at snakke om problemet would also sound natural.

Very roughly:

• tale can sound a little more neutral or slightly more formal
• snakke can sound a little more informal

But in normal conversation, both are common.

It is grammatically in the present tense:

• bliver
• tale
• bliver

But like English, Danish present tense can describe something happening now, a general situation, or even a future plan depending on context.

So this sentence can mean something like:

• We keep talking about the problem until we agree.
• We will keep talking about the problem until we agree.

The exact time often comes from context.

In modern Danish, this depends on comma style.

With start comma, you write a comma before a subordinate clause:

• Vi bliver ved med at tale om problemet, indtil vi bliver enige.

Without start comma, you can write:

• Vi bliver ved med at tale om problemet indtil vi bliver enige.

So the comma in your sentence is correct, but you may also see the sentence without it.

Normally, no. You would usually just say:

• indtil vi bliver enige

not

• indtil at vi bliver enige

For a learner, the safe rule is: use indtil directly with the clause.

In careful speech, bliver is something like BLI-ver, but in everyday spoken Danish it is often reduced.

Many learners notice that it may sound closer to:

• bliver with a very soft second syllable
• sometimes almost like bli’r in casual speech

That kind of reduction is very common in spoken Danish, especially with frequent verbs.

So if you do not hear every sound clearly, that is normal.

Yes: it has a very common Danish pattern:

• main clause + indtil + subordinate clause

And inside the main clause, it uses another very common structure:

• blive ved med at + infinitive

So this sentence is useful because it combines two high-frequency patterns:

1. Vi bliver ved med at ... = We keep ...
2. ... indtil vi bliver enige = ... until we agree / reach agreement

If you learn those two patterns, you can make many similar sentences:

• Jeg bliver ved med at læse, indtil jeg forstår det.
• De bliver ved med at diskutere, indtil de bliver enige.