I get that Ukrainian does not use the definite and indefinite articles, but it seems like sometimes "a/an" is implied somehow, and sometimes "the" is implied somehow. How would I know, in this case, that "This is A center" is correct, and not "This is THE center?" How would "This is the center" be stated differently?
Translationwise both are correct! There is no "how would I know" because there is no concept of articles in Ukrainian, so the language itself intrinsically doesn't "know" whether it's "the" or "a" :) If either translation is not accepted, report it.
For us, "центр" doesn't mean both "the center" and "a center", it means neither because we just don't think in terms of "the" and "a". If we want to get a feeling or an effect similar to "the" in English, we say "that center" or "this center" or directly "center of [sth]", and for the flavour of "a" we would say "some center" or "any center" or "one center".
Very strange. Are you sure your input did not contain any other mistakes or was not misspelled? Please, whenever you write a comment, quote your input.
I just checked the exercise, and "center", "center" and "downtown" are all accepted. The last modification of the exercise happened 6 months ago, so these have been accepted all this time.