unsolved

Unsolved problems are problems for which we neither have a solution, nor know whether such solution can be found.
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solvable

P is solvable if, at some point in time, a proof will be provided that solution to P can be found.

unsolvable

P is unsolvable if, at some point in time, a proof will be provided that P does not have solution.
illustration of
illustration of "unsolvable and undecidable problem which leads to madness", black background

fatal

Problem P is "fatal" if no proof could ever be provided whether P is solvable or unsolvable. 

It is impossible to tell about problem if it is "fatal". That's why it is fatal ;)