If codewars is like an open book test, then taking a solution you found and applying it wouldn't be cheating. Still, it doesn't seem like someone who copy/pastes a solution was the one who solved the problem.

There are at least three sorts of cases:

1. Learning about an algorithm (from, say, Wikipedia) but taking no runnable code from elsewhere.
2. @elasolova's cases where the handful of ways of writing something are somewhat widely known or at least widely discussed. (Everyone's Sieve of Eratosthenes looks quite a bit like everyone else's.) Background knowledge -- or what should be beackground knowledge in a given domain -- never needs to be sourced.
3. Copying an entire solution without sourcing it.

3's that aren't 2's get you yelled at on stackoverflow, expelled from school, or sued, but there's no simple rule distinguishing 3's from 2's. What counts as common knowledge is up to a given community. I have a couple times considered posting links when I found this kind of bald plagiarism, but this community's norms -- if they exist at all -- are not as clear as, say, SO's.

The only forms of cheating are bypassing the tests, and "miraculously" ending up with a solution identical to another user's one when that'd be impossible.

Anything else is not deemed cheating.

So, I have been wondering. Is it ok for us to use code from internet for some katas as long as we give the source? Because some of them are mainstream problems and there are already well known solutions not to be written from scratch. Or does this also count as cheating?

Yes happened to me also in a similar test case. (Java)