How long is the language course? If it's relatively long, then "live" makes sense. I.e., the longer the amount of time, the more one would tend to use "live" instead of "stay". I would not say they are synonyms, of course it always depends on context!

Sybille: Your first thoughts were right. To live somewhere generally implies some degree of permanence. Normally, people do not "live" in hotels. As you say, people generally "stay" at hotels. "Live with a family" is, I think, sometimes used in advertising texts. It conjures up the idea of becoming a temporary "member" of the family.

Usage varies around the world. I once met a girl from Singapore who said "We stay in a house" (i.e. not a flat).

@3: Well, if that was the entire sentence, didn't she get the tense wrong as well? ;-)

Note that in Scotland, stay is commonly used to mean live, e.g.:

Q: Where do you stay?
A: In Glasgow

In England (and elsewhere) this would have to be: "Where do you live?" (or "Where are you staying?" if the context is perceived as limited in time / not permanent), unless the questioner is asking about A's normal accommodation when A is away on business trips, for example.

for one thing, people can also live in hotels; lots of them do, in fact.

secondly, I think if your life takes place in mostly one particular environment, especially in a familial environment (and you're staying with this family for longer than say a week), there's nothing wrong IMO with saying "live" for that period.

@6: Yes, some people live in hotels. Some famous people, I believe, have lived for years in hotels. Where I live, some homeless families live in hotels at the expense of the state. Hotels are, in my opinion, not normal permanent housing for the overwhelming majority of people. Therefore, I would say that the overwhelming majority of people do not "live" in hotels. When you say "lots of them do, in fact," how many would that be in your experience?

@SD3

No, her tense was right. She meant what I would mean by "We live in a house."

@SD3: Having lived in a hotel myself once - in the distant past - and having known quite a few people at that time who also lived in hotels, I'd say the number of those people is probably a lot higher than the general public would assume. I have no statistics to back that up, of course, so I can only offer you my personal experience, limited though it may be.

#8: Yes, I assumed that was what you meant. My comment was intended to be humorous - I even remembered to add a smiley!