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