This problem is commonly solved in a somewhat more meritocratic fashion by having entrance exams. My alma mater had them (not in the US), and problems in the ex

This problem is commonly solved in a somewhat more meritocratic fashion by having entrance exams. My alma mater had them (not in the US), and problems in the exams spanned a range from "quite hard" to "insane". You were not expected to solve everything. Then they'd grade the results anonymously, and pick the top N, where N was capacity for that year. To have a chance of passing the exam you could sign up for their prep school for a very nominal fee that only covered postage (it was by mail), and many of us did.