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<p>The reason that schools like MIT and Caltech are considered as schools for “geniuses” is that they are mostly focused on math & science, whereas schools like Columbia and HYPS are much more well-rounded. Usually, people tend to think of geniuses as those who have talent and passion in math & science, so I think your comparison here is not applicable.</p>
<p>Also, in schools like Columbia, there are plenty of “geniuses” in their own rights, just that they don’t confine themselves to math & science but rather seem to have wider interests.</p>