Is it a must to have the underlined definite articles placed in the sentences below? Could you comment on why or why not? Also in 2 & 3, assuming the answer is "yes," must they be capitalized?

1. Come to Seoul, a city where the old meets the new. (This is a tourism organization's motto referring to old and new cultures).

2. Please send your advertisement to the Korea Times, Chicago branch at the address shown below.

3. I belong to the Korean-American Scientists Association.

Thanks in advance.
I'm not a grammarian, but I believe that 'the' in no.2 is unnecessary, unless the name of the [newspaper?] is The Korea Times. In addition, it seems the instruction in no.2 is unclear as written: is the mailing address 'Korea Times, Chicago branch' (at the address below), or 'Korea Times' (Chicago branch at the address below)?

I would tend to keep 'the' in no.3, but capitalized only if the name of the organization was The Korean-American Scientists Association. If the name of the organization was Korean-American Scientists, I'd write, 'I am a member of Korean-American Scientists'.
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Thanks a lot davkett. I still have some more questions though.

Now if you were naming the Korean-American organization, would you name it The Korean-American Scientists Association or just Korean-American Scientists Association? Which one sounds more natural?

As far as (the) Korea Times, yes it is the name of the newspaper. But I've seen it written both ways, with and without "the." But most, if not all, of the newspapers in the U.S. include "the" as part of the name as in "the New York Times" or the "the LA Times," I believe. Is there a reason? Plus you don't have to capitalize it, right?

And how about the United States of America? You don't have to capitalize "the" here, but isn't it part of the name???
If 'the' is part of the name, it is capitalized; but I'm not convinced that most newspapers have 'the' as part of their name.

In naming the organization 'Korean-American Scientists Association', the acronym would be KASA. With 'The' in the name, I suppose it would have to be an unpronounceable TKASA.

I think, in speaking, it sounds more natural to say 'the' before any named organization that doesn't have 'the' in the official name. The implication in saying the is 'the organization called' [KASA].

As for 'the United States of America', we always see the abbreviation-- 'the USA' (not 'The USA').
Thanks a lot davkett for your very helpful answers!!!
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Let me squeeze in and ask, "why a public organization with a known name like Immigration and Naturalization Service (I.N.S.) would have "the" in front when starting an article?

Starting an article -

The Immigration and Naturalization Service will institute some new measures to curb illegal entrants.

The name of this public entity is I.N.S., not "the I.N.S." but why "the " in front when starting an article?