Many of these problems are indeed quite elegant, but way too many (generally the geometric ones) require the use of theorems that even many math majors may not know — this is more for the "contest math" genius in high school more than necesasrily a math major, and even they may not know the theorems required.
And not all of these problems are necessarily "quickies". For instance, one asks multiplcation of a very large number by 125 — the quickie solution is to multiply by 1000 and divide by 8, but this is not even so "quick".