I competed in both of these, and in Mayhem I was fortunate enough to win Path of exile currency two prizes to being in the top 5 of two classes. The adventure of pushing to get a win (also known as a Demi, following the decoration, Demigod's Dominance) was definitely a grueling one, also took a considerable amount of practice. If you want to go for a Demi of your own, here are some tips for racing in Path of Exile which should help.

This is easily the main tip. If you are going to try to get a Demi, you are going to get to put in a lot of hours. If you have other obligations during the time period, or can not play extensively because of your work schedule, you're likely to be hard pushed to win. Not to say it's impossible, but it'll be very hard. There is nothing worse than coming first position and simply missing a Demi since you couldn't put in enough hours.

If your plan is to go for a Demi but can not or don't need to play 24/7, do not play Softcore. The top five Path of Exile players from each course in each league get Demis. The Softcore Path of Exile races don't have permadeath, and as a result, they are a total grindfest. The people who can grind the most will win each and every time. I knew I wouldn't be able to commit to this amount of play time, therefore I picked the Hardcore leagues instead.

On a similar note, it's worth avoiding the 'favorite' classes. Part of the reason I managed to win 2 Demis in the exact same race was since I picked two quite unpopular classes--I did not need to buy poe orbs compete against teams, also that I managed to move up the ladder quickly.