Which condition must be true for a nation to exist?

There isn't a single condition.
You could say you need to have passports and own laws.

Another big theory is, other countries need to accept you as a nation. A big example of a country who doesn't have this is the country within georgia (the country not state) they have laws, passports but Europe and the US have not acknowladged its existance

Ithink it’s a : ) good luck on whatever