Name A Problem That Nationalism Has Solved
None. Nationalism creates an "us" and "them"
Nonsense. "Us" and "them" preceded nationalism by several thousand years. Before we had nations, we had city-states, before that we had tribes. Medieval Europe wasn't all that great a place to live.
It has spread the scourge of "the Limbo" worldwide.
Things Nationalism has solved:
- Tribalism: Instead of small bands of blood related people fighting against other small bands of blood related people in battles that are waged purely for genetic reasons - nationalism instead allows individuals unrelated by blood or marriage to live and work together.
- Nomadic Life Style: Instead of having to wander and completely live off the land nationalism allows unrelated people to build civilizations (agriculture, governments, academic institutes, i-phones, etc).
- Ceremony and Identity: Nationalism promotes a sense of community by common practices and beliefs while also providing individuals a clear understanding of their place within their communities and within their broader worlds.
. . .
Nationalism - in and of its self - isn't necessarily a bad thing. And quite often it's pros far outweigh it's cons. However, it can be dangerous when used as an "us vs. them" pretext (which, unfortunately, does happen).
I think your question is a little vague and general. I'd say it depends of what you mean by nationalism, what you mean by solving, and what you mean by problem. And of course, it also depends on which side of the problem or solution you are.
"Us" and "them" preceded nationalism by several thousand years.
I have heard this explained as being the result of how the human species originally evolved in small groups, which (to me, at least!) seems feasible. The "Us" and "Them" attitude is still very much alive and well; one only has to look at sports teams - and, perhaps even more significantly - their supporters to see evidence of this.
One achievement of nationalism was to at least lift this "Us" and "Them" tendency from out of the parochial level.
(Others here have correctly also identified what has sometimes been termed "ultra-nationalism", and which is one of the building blocks of Fascism).