I graduated from Harvard Med with honors, so I get the whole academic competitiveness thing. Frankly, I have always enjoyed academic competition and have thrived on it. I've tried to pass that on to my kids, who are extremely uncompetitive.
I just can't see making your child (or yourself) miserable over all of this. It just isn't worth it. How many such kids get into the Ivies, etc. and fail? They must face up to the fact that they aren't as gifted/smart/whatever as their parents want them to be.
The Chinese moms in my son's third grade class have all told me approvingly that he is the smartest kid in the class. I don't know if he is but I guess I'm glad that they think so!