1.- There are 3 quake countries here: Chile, Argentina and Brazil, in that order. The rest simply don't exist at all, as can be checked in qlranks.
2.- Out of those, the only one that has constant tournaments and cups is Chile. The ratio is something like 10(CL)/1(ARG)/0.1(BR), so I'll leave the former for next.
3.- There are very few BR clans (actually I know of just one) and the Argentinian clans (lets say 6) practically don't play friendlies between each other or with Chileans; they are just a tag with at most 2 exceptions (kc and eb) that still don't play frequently as a team. The sole BR clan has not played friendlies or tournaments in more than a year if not 2, so disregard it completely from now on.
4.- On Argentina 99% of team gametypes activity is CA, with 99% of it being pubs. Not exaggerating, they simply do not play and do not know how to play ctf (we literally beat their best team 8-0 8-0 8-0. Stream vod is somewhere) or tdm (that's even sadder because they lose that by WO lol, but if they show up is the same story).
5.- Argentinians and Brazilians don't like playing with each other or with other countries. I know it sounds bizarre but it's true, with plenty of examples. The latter is both started banning each others IPs from their "offline"/pirate servers, and when Argentinians still had official servers they banned Chileans from playing their tournaments (yep).
6.- Players from those 2 countries cry about pings to the point they think 40=unplayable and try nothing to fix it. There's exactly one argentinian clan that doesn't (called kc) and very few individual br players (one is flesh1).
7.- In the end Argentinians and Brazilians clans simply aren't interested on playing on a nations level, whether as countries or as clans.

So that's the situation of team modes in 3-1 world countries. Argentina and Brazil simply don't exist on a southamerican level, especially Brazil, with the sole exception of 1 argentinian clan that still isn't very active as a team. There you go, one team that isn't from Chile, just one.
On the national level, Argentinians rarely play friendlies, and when they do it's ca and only with each other. Brazilians don't even do that.

I know the first reaction is "but why Chileans don't sign up to Argentinian tournaments. Problem solved", but thing is Argentinians practically don't host tournaments (not even ca. At most it's been 4 since the beginning of QL) and when they do they do not allow chileans nor brazilians on them.
Another problem aroused last month: Argentinians servers were shut down, therefore if they ever organize another cup it will have to be played using the offline client. I write this because another obvious reaction is "why they don't change servers to have similar pings", but people simply didn't in all QL's history and now it is impossible without going to the unofficial servers.

tl;dr: Since there are QL servers here in 2009, there have been exactly 0 nations cups or proper southamerican clan tournaments. With very rare Argentinian tournaments and 0 from BR, it has always been about Chilean tournaments in which argentinians and brazilians are welcomed and encouraged to participate, but they rarely do. Lets say I remember a br clan joining our tournaments twice, same for argentinians and in both cases it is the same clan (the said inactive br clan and kc).

Now comes Chile. To do a sum up of the quake activity since the 90s, here it's been a lot more developed than in other SA countries, with more players, bigger prizes and so on (Remember were the cpl pk stop went to) until the bubble burst around 2005.
Also it is worth noting for some reason quake2 was always bigger than quake3 until it completely died around 2007, who knows why on both ends. Then before ql q3 was started to die fast, so with both quake3 and quake2 on trouble QL was released on the perfect moment which is why Chile had one of the very first locations around the world.

Anyway, now on Q3 Chile always had more activity in mods with items than mods that don't have it, but in QL it started decaying to the point public play is a CA-fest as well. There's still some ctf activity going on, but mostly clan matches. TDM is practically non existent, with 0 public play, 0 draft pickups (0) and some sporadic tournaments but literally all of them were lackluster (as I say, if I win tournaments there's something seriously wrong with the community).
In short, here it's 95% CA, 4% CTF and 1% TDM, and decent players share my point of view of laughing at ca but their sole alternative is ctf. Think of it as the Argentinian situation (public activity is all ca) but with people actually playing clan matches every day instead of once every month, and 25% are on ctf.

Up to 2011 there were plenty of tournaments here, but nothing like it was on the good old quake2 days (I remember tournaments with 64 clans) or with huge prizes. Those were mainly CA and CTF, with some lans, drama and whatnot, fun times.

Problem is in mid 2011 and onwards the competitive scene literally died, with server hosting problems at the same time. CTF was turning into an unreliable alternative, with huge amounts of WO (50%), and people that volunteered to be admins didn't care enough to do all the effort and disbursements to finance CA.

tl;dr: They are now trying to revive it (CTF of course) but I'd say it is going to have only 12 clans with about 33% of WOs, whereas 2 years ago It'd have been 24 clans with 10%.
0 BR clans (obviously), 1 argentinian clan (kc) that is trying to get friendlies going with other argentinian clans but they simply can't find any.

I've posted repeatedly that I'm not much of a duel player and that I have little experience on it on the national and southamerican scene (lets say I'm the type of player that shafts his way through rankings. Now I should be around 1800-1900 skill level heartless style). With that said, here are my appreciations.

Chile has always been a clan based nation, with duel being on the side. Lets put it this way: sure there are some good ones here but team gametypes are the ones featured on lans.
There are some duel players (lets say 30 total) and some of those are pretty dedicated, including organizing short online cups and such. Problem is they started to retire, which is a shame considering some were actually talented and not just 24/7 practice making them better than the rest.

As I said, Brazil is practically non existent in quake. That's not because of lack of players but lack of a decent medium skill level (it is very, very, very low) and most importantly lack of organization, which is the reason why they never had an official server and never bothered to create clans, with the said one exception that is inactive.
Now, the thing is they have some very good duelers. It's like 99.9% of BRs are completely awful but 0.1% excels vastly at it. There are lets say 8 duel players, and the best one is flesh1 which is the best southamerican dueler. They mostly play with each other (offline server) but some of those brave 8 play on Chilean servers from time to time (the only official server left here. Argentinian was closed last month).

Argentina is a pretty interesting case. 95% of players just play CA-only (please search "rad" on quakelive.com), but the other 5% do that plus duel. Absolutely 0 play tdm or ctf, but they play duel. I'd say they have about the same amount of duelers as chile if not more, which is pretty good. Their biggest exponent is doomsday, their legend is pipi (went to quakecon, didn't make top8, excuses).

And now come the problems. From the outside it looks like a healthy playerbase, but in Chile duel was secondary, BRs practically don't exist with the exception of few players and Argentinians don't compete. In the end it is a huge lack of organization + people not wanting to play in each others tournaments.
With that said, it's the same as the clans situation: mostly chilean tournaments with argentinians and brazilians accepted to play here.

Now, I remember 2 exceptions that can be called southamerican tournaments: one that I only saw afterwards because someone posted it here on esr lol (won by chilean etjx, second place doomsday as far as I remember) and in 2012 4 jeesports one day cups that ended up in a final 5th, all online, organized and played on argentinian servers. I'd say the latter is the only proper southamerican tournament, and about 80% of players were argentinians nonetheless.
First and second qualies were won by flesh1 afaik, who stopped playing onwards because of connection issues (that BRs recently solved using vpns, finally). On the 5th and final cup the 3 first places were chileans (the only 3 that participated).

tl;dr: Absolutely 0 southamerican duel lans, no further southamerican online duel cups being organized or thought about, only one ever done and that's why it was played on Argentinian servers otherwise Argentinians simply don't play on Chilean servers tournaments.