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I was also glad that the topic of competition came up in this discussion. For me, sports were never a place of competition between people and instead they were a place of competition with myself. That being said, I usually stuck to team sports and when I did try individual sports I did sense more competition between individuals and so I promptly quit those sports. It was always drilled into my head that in team sports you "win as a team and lose as a team." This made me feel like I had less responsibility on my shoulders and I could improve my abilities simply because I wanted to, not because I had to beat the individuals on the other team. Sure, I was happy when we won, but it wasnt the end of the world (most of the time..) when we lost. Where I did feel the pressure of competition between people was in the classroom. In my orchestra classroom we consistently had "Challenge" days where we were forced to challenge other individuals in the class for higher seats. I found these Challenge days to be horrible. Not only was I being compared to other people purely on a single ability, it was also being done publicly and it had a grade tied to it. Supposedly the orchestra was a "team" of sorts, but I really dont believe it. We might have been working together to create a single sound but it was the one place in my life that I felt the greatest amount of competition between people. 

I dont think competition needs to be eliminated in school, it just needs to be redirected. Critical thinking and learning to learn are a form of internal competition. In learning to learn and critical thinking, you are pushing yourself to go beyond what you already know or think. If competition is eliminated from school completely, you will end up with a school full of students who are indifferent to the educational process.