Solving problems isn't an obvious good, or at least it shouldn't be. There are in fact bad problems.
For example, MKUltra tried to solve a problem: "How can I manipulate my fellow man?" That problem still exists today, and you bet AI is being employed to try to solve it.
Some not-problems, presented as though they are:
"How can we prevent the untimely eradication of Polio?"
"How can we prevent bot network operators from being unfairly excluded from online political discussions?"
"How can we enable context-and-content-unaware text generation mechanisms to propagate throughout society?"