Learn Polish: Mieszkam w centrum miasta. - I live in the city centre.

Normally, the preposition w (in) makes the following noun take the Locative case. However, centrum is a special foreign loanword. In the singular, it is indeclinable, meaning it stays exactly the same no matter the grammatical case.

Why is it miasta instead of the dictionary form miasto?

Miasta is in the Genitive case, which is the Polish way of expressing possession or the word 'of'. Literally, the phrase means 'in the centre OF the city'. For neuter nouns ending in -o, like miasto, the Genitive ending changes to -a.

Can I use the verb żyć instead of mieszkam to say 'I live' here?

No. Polish has two different verbs for 'to live'. Mieszkać (used here as mieszkam) specifically means 'to reside' or 'to have your home somewhere'. The verb żyć means 'to be alive' or refers to life in a broader, more general sense.

Do I always need to add miasta when talking about the city centre?

Not always! If the context is clear, you can simply say Mieszkam w centrum (I live downtown / in the centre), which is very common in everyday conversation. Adding miasta just makes it slightly more formal or specific.