Princeton? Brown has very good Applied Math program.

数学背景强特适合CMU。

游泳只参加JV的话，对申请影响几乎没有，甚至Negative 影响。Same applies to leadership position. If VP of class is for junior and senior, big plus. Otherwise it's not really strong. 

Is you son interested in finance? Univ. of Chicago is also a good choice.