Pure mathematicians just love to try unsolved problems - they love a challenge.
Just because we can't find a solution it doesn't mean that there isn't one.
It's fine to work on any problem, so long as it generates interesting mathematics along the way - even if you don't solve it at the end of the day.
I had this rare privilege of being able to pursue in my adult life, what had been my childhood dream.
But the best problem I ever found, I found in my local public library.
The greatest problem for mathematicians now is probably the Riemann Hypothesis.