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In 1939, George Dantzig, a graduate student at UC Berkeley, solved two open problems in statistical theory, which he had mistaken for homework after arriving late to a lecture by Jerzy Neyman. The two problems ended up being his entire PhD thesis.pic.twitter.com/0K6wkDgey0
On another note, I used to have a professor who sneaked unsolved problems in worksheets. I remember spending hours on one of them, finally googling it, and discovering that it was unsolved. Bad memories.
Hopefully it gives you an insight of how France got their Fields medals 
They are only thought to be unsolvable, maybe we only lack the knowledge of an method to solve it.
Why bad? That made you think...
I understand that the book of functional analysis by Rudin had a lot of "research problems" in. He set them as homework, apparently it was his busiest time!!!
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