Upvote:1
You can enter/leave the US on your American passport and enter/leave Brazil on your Cyprian passport. As long as you're entering and leaving a given country on the passport of the local country (if you have one) or the same passport (if you don't have a local one) you should be fine.
In general airlines will not check your visas for you (one exception I have found is Porter Airlines from Toronto to the US, presumably because they found the cost of repatriating people who didn't sign up for ESTA exorbitant).