I will start by taking you back almost 35 years, when I was a young schoolboy. My parents always told me that there are only four positions in the class: first, second, third and "the rest". So naturally I was all the time being pushed to try and leave "the rest" behind and and get into the first three. I did manage to do that eventually, but there was always the these other two guys around me trying their best to get or stay ahead of me. Then, there always were boys from "the rest" of the crowd threatening to break into the top three and often succeeded.
Then something happened that got me to the first position and I never looked back. That something was when I stopped competing with the others. I decided that if I have to stay on top I must achieve a certain percentage and that was as close to 100% as possible, and I stopped worrying about what anyone else was doing or achieving. It was just me trying to get to 100%. In some subjects it was possible to get to 100% while in others you could only get as far as the teacher was willing to let you go. So now I had a goal in each subject to achieve and that goal was very precise and clear. It wasn't to stay ahead of others, as that would be a moving target without any specific position to get to. I was very clear what I needed to get in order to be the best. I repeated the same when I was in the engineering school and got similar results.
What I am trying to explain here is that if you want to be the best in whatever you do, you have to forget about how the others are doing and set up your own target, and then focus just to get to that target. The end results will always be that since the target was set at a point where you would be the best so you will turn out to be the best. This is what I consider not competing with others, or in other words, competing with yourself.