> And simply knowing a problem can be solved is half the battle.

Have you done any mathematical research? If not, then no, knowing that a problem is solvable is not “half the battle”.

Homework problems are all designed to be solvable, yet they can vary greatly in difficulty. Research mathematics is even more extreme, because, unlike with homework, you don’t know that it is solvable with the extant mathematics, and you might need to invent new maths.