Αυτό το πρόβλημα με απασχολεί.

In Greek, a demonstrative like αυτός, αυτή, αυτό normally goes together with the definite article when it means this or that before a noun.

So Greek says:

• αυτό το πρόβλημα = this problem
• literally, something like this the problem

You normally would not say αυτό πρόβλημα.

A few useful patterns:

• αυτό το πρόβλημα = the most common neutral way to say this problem
• το πρόβλημα αυτό = also possible, often a bit more formal or emphatic

So the article το is not extra by mistake; it is the normal Greek structure.

με means me.

It is the weak object pronoun, and it comes before the verb in normal statements:

• με απασχολεί = concerns me / occupies me / is on my mind

English puts me after the verb, but Greek usually puts this short pronoun before it.

Compare:

• Το ξέρω. = I know it.
• Με βλέπει. = He/She sees me.
• Αυτό το πρόβλημα με απασχολεί. = This problem concerns me.

So με is the person affected by the problem.

It is the subject.

That can be confusing because in Greek, many neuter nouns have the same form in the nominative and accusative. So το πρόβλημα can look the same whether it is subject or object.

In this sentence:

• Αυτό το πρόβλημα = the thing doing the action of concerning
• με = the object, the person being concerned

So the structure is:

• subject: Αυτό το πρόβλημα
• object: με
• verb: απασχολεί

In English, the meaning is like This problem concerns me, where this problem is also the subject.

Because the subject is this problem, which is third person singular.

The verb is απασχολώ in the dictionary form, meaning things like:

• to occupy
• to concern
• to keep busy
• sometimes to trouble or to preoccupy

Its present-tense forms include:

• εγώ απασχολώ = I occupy / concern
• αυτός/αυτή/αυτό απασχολεί = he/she/it occupies / concerns

Since το πρόβλημα = it, Greek uses απασχολεί.

So:

• Αυτό το πρόβλημα με απασχολεί = This problem concerns me
• not απασχολώ, because the subject is not I

It can cover a range of meanings, and context decides the best English translation.

Common senses of απασχολώ / απασχολεί:

• occupy
• keep busy
• concern
• preoccupy
• sometimes trouble

In this sentence, it usually means something like:

• This problem is occupying my mind
• This problem concerns me
• This problem is troubling me
• This problem is on my mind

It does not always mean strong emotional worry. It can simply mean that something is an issue you are thinking about seriously.

So με απασχολεί is often broader than just worries me.

Yes. Greek word order is more flexible than English.

The sentence you have:

• Αυτό το πρόβλημα με απασχολεί.

is perfectly natural.

But you could also hear:

• Με απασχολεί αυτό το πρόβλημα.
• Αυτό το πρόβλημα απασχολεί εμένα.
  (more emphasis on me)
• Εμένα με απασχολεί αυτό το πρόβλημα.
  (stronger contrast or emphasis)

What usually stays true is that the weak pronoun με comes before the finite verb in a normal statement.

So the word order can change for emphasis, but the original sentence is a standard, neutral pattern.

Because πρόβλημα is a neuter noun.

Greek demonstratives must agree with the noun in:

• gender
• number
• case

Since πρόβλημα is:

• singular
• neuter

the demonstrative is also neuter singular:

• αυτό το πρόβλημα

Compare:

• αυτός ο άνθρωπος = this man
  masculine
• αυτή η ιδέα = this idea
  feminine
• αυτό το πρόβλημα = this problem
  neuter

So αυτό is used because of the noun’s grammatical gender, not because the thing is somehow especially abstract.

With many neuter nouns, the nominative and accusative forms are identical.

For το πρόβλημα:

• nominative singular: το πρόβλημα
• accusative singular: το πρόβλημα

So form alone does not tell you the case.

You figure it out from the sentence structure:

• με is clearly the object pronoun = me
• the verb is third person singular
• therefore Αυτό το πρόβλημα is the subject

This is very common in Greek with neuter nouns, so it is something learners get used to quickly.

Yes.

• Το πρόβλημα με απασχολεί. = The problem concerns me.
• Αυτό το πρόβλημα με απασχολεί. = This problem concerns me.

Adding αυτό makes it specifically this problem, not just the problem.

So the difference is the same basic one as in English:

• the problem
• this problem

If the context already makes it obvious which problem you mean, either might be possible depending on what you want to emphasize.

Yes, very much so.

Greek speakers often use με απασχολεί for things that matter to them, are on their mind, or need attention.

For example:

• Με απασχολεί το μέλλον μου. = My future is on my mind / concerns me.
• Με απασχολεί πολύ αυτό το θέμα. = This issue concerns me a lot.
• Τι σε απασχολεί; = What’s bothering you? / What’s on your mind?

So this sentence uses a very natural and useful Greek pattern.