[vouaobrasil]

> It's a shame the school system doesn't put more emphasis on why you'd need to solve such difficult problems in the real world, as in, what are their theoretical or practical applications of the knowledge gained from solving such problems.

Speaking as a PhD in math, Olympiad problems are very different than ordinary research math problems. Olympiad math problems in particular don't require creating new theory, but rather brilliance and knowing a bunch of standard tricks and how to apply them in insightful ways.

Becoming good at solving Olympiad problems won't really give you much skill in solving creative research problems, but it does give some indication of pure ingenuity as one dimension of mathematical intelligence.

[posed]

I don’t think that is cut and dry as such, being able to solve IMO(International Mathematics Olympiad) level problems require a significant level of creativity and imagination, and folks who are good at it naturally have an aptitude(and perhaps an upper-hand as well) for problem solving in general, research or otherwise.