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My younger son gets upset when his older brother beats him at anything!
From what you say, his competitiveness may be limited to the situation at home, and it will not necessarily spread to school. Some children show far more intense sibling rivalry than others do, and it sounds as though your son is experiencing it in spades. Although the competitiveness appears to be mainly directed at his older brother, the birth of the younger sibling may actually have triggered or intensified it. And since the baby is a girl in a family that didn't already have one, it may make things even harder for him.

Being a middle child is not an easy task (at least so my husband, who is one, asserts). The older child can do more things, and the younger one gets more attention. Thus your task is to try to balance the situation for your middle son. Avoid situations in which he has to compete with his older brother, as you say you are now doing. Help him find and interact with friends his age outside the family, and do what you can to nourish interests that don't coincide with his older brother's. And find time to give him some of the special attention he seems to feel his baby sister has taken from him. These things are not easy to do; it requires a lot of juggling on the part of parents. But it can be done. Good luck.
Dr. Bettye M. Caldwell Ph.D. Professor of Pediatrics in Child Development and Education