All you need to do is play 5 online games to participate in tournaments, as for team matches, check out my profile, lookinto Venom Chess or Chess Shinigami. If interested in either one, seek membership through I and I'll get you in. Once on board, you can play any team match that it's available. If there isn't any team match available for your to participate in; then, I'll make one for you.
Tournaments and team matches
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I've been using this site regularly for the past month and I have been attempting (to no avail) to participate in a tournament or team match.
I have a couple of ideas to make it easier for people to actually participate.
First of all, regarding the fact that non-premium members are only allowed to be in one tournament at a time- Instead of setting it up so you can't sign up for one if you are signed up for another one, it should be set up so you can't sign up for one if you are actually playing in one. This way a person won't be waiting for weeks for the one tournament to start, and then find out that their rating changes during that time caused them to be removed, so they have to start all over completely and wait a few more weeks.
Second, regarding team matches- Say you are rated at 1585, and your team has two open team matches one for <1600, and one for >1600, you sign up for the <1600, and while you are waiting for the match to start, your rating goes to 1605. Now you're removed when the match starts. If you'd tried to be added to the >1600 match when you signed up, you would have just gotten a message saying you can't sign up because your rating is too low. Obviously you see the problem here, the person in this scenario doesn't play in either because their rating is too close to the borderline. My suggestion is that it should be set up so anyone can sign up for any team match, but only people in the correct ratings range at the time it starts will actually play. Either that or base it all on each person's rating at the moment they sign up instead of the moment the match begins.