Depends on what you call a solution. If you call there is no such function a solution, then yes every well posed problem has a solution.

But many people call only an object that exists a solution and has no solution just means no such object exist.

If you take ill posed problems - the barber that shaves everyone that doesn’t shave themselves. Then you are no longer able to answer with true/false so in the limited context there is indeed no solution.

Plenty of problems have no solution, even in PDEs land where we're quite willing to redefine what's a solution (weak solutions, distributions etc.)

What is the solution for avoiding death and taxes?

seven bridges?