-first sentence: good (although I didn’t realize that Christ’s Hospital was not a literal hospital, so that threw me off for a minute 
I’m guessing the adcoms won’t have that issue)
-second sentence: I’m not sure about the negative tone you set up with “adolescent and aggressive.” It strikes me as being very judgmental, in a way that isn’t negated by the “caring and familial” in the second half. I don’t know if that’s the atmosphere you want to build in 100 words.
-third sentence: hmm. You kind of lose me here. The lack of elaboration on “the extra curricular activities” (extracurricular can be one word) makes that part feel out of place. And the “However” with which you start off the fourth sentence isn’t working for me either – you don’t seem to be contrasting anything. Finally, the “competition” that you reference doesn’t seem like the sort of thing that would impress adcoms in a positive light. Brown doesn’t pride itself on being cutthroat; the environment here is much more collaborative than that. There’s nothing wrong with ambition, but I could see how saying that you want to “emulate competition” at Brown could potentially strike adcoms in a negative way.
As a disclaimer, I have nothing to do with admissions. But my gut feeling is to suggest that you elaborate on the positive ways that Christ’s Hospital has affected you – how discussions with your peers have shaped you, in what ways have you become more flexible, how you want to channel your ambition, etc.