Maybe it's because I'm a girl, but I think Arunangelo has a point. Granted that there are some aspects of competition which are built into our culture, but most of the competition we see is that which attempts to deny another person what we want for ourselves, and didn't Jesus say something about loving our neighbor as we love ourselves?
Paul speaks of running the race to get the prize. This is more, the way he is using it, competition against oneself so that one does not get weighted down by the world and its temptations.
And yes, we are called to be servants. One of the things which drives me nuts is the common advertisement for Christian schools and such that says "We make Christian leaders" or words to that effect.
God will choose the leaders HE want. We should be making Christian servants. We should be teaching obedience to the Lord first and service to others as well (which is often the same thing). And this does imply a rather stark lack of competition in many, many areas of life.