Do non-Russian technical universities use Olympiad-style math/problem solving entrance exams? 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.