In the U.S., colleges typically don't use entrance exams to select students. For test results they rely instead on standardized tests like the SAT or ACT. I heard that the top technical math/science schools in Russia administer entrance exams which are similar to math Olympiads. I am wondering: is this a Russian thing or do universities in other countries use difficult problem-solving/Olympiad tests like this?
In Japan, many universities have their own entrance exams. For the entrance exams for mathematics that I've seen, there are several challenging problems (maybe about 5) to be solved in about 2-3 hours. For instance, here is a recent exam from Tokyo University (this is the math exam for science students--humanities students get a different exam). If you're a bit rusty on your Japanese, it's 2.5 hours for 6 problems.
Entrance exams in other east Asian countries are reputedly difficult also, but I haven't seen those exams to compare. Maybe someone else can comment on that.
Another bit of anedoctal evidence from my experience: in Italy, the Scuola Normale Superiore (a very highly-rated honors boarding college for the hard sciences and the humanities) uses Olympiad-like problems for its admission tests in mathematics and physics for bachelor and master students.
(All problems in the links are in Italian, sorry for that).
Actually, today's Russian entrance exams are quite as standardized as in many places abroad. Besides this, state exam in mathematics not only contains the part, filled with test questions, but also staffed with an extra one, where contestants are required to write down all the solutions in details. It's not an obligatory - it's possible to pass the exam without solving this part, but it's impossible to gain a result, which would be sufficient to be accepted to the undergraduate programs in technical universities.
Moreover, regarding olympiads, for the last years some universities tend to organise their own olympiads, results of which are always taken into accout in the process of application. In most cases, the contests' problems looks like classical entrance exam ones, which inherited traditions from the past times. Although, some olympiads enable contestants to solve difficult problems( IMO style), which require a strong background and sufficient level of training. (such as Moscow Mathematical Olympiad or the Russian Mathematical Olympiad).