When managers ask for something they are typically not asking a math question. They are asking for a solution to a real world problem. There are always approximations of a solution even if a perfect one doesn't exist.

Yeah I think this one is just semantics/being at least a bit politically aware.

"There is no solution" is for most shorthand for "I can give you 20 solutions, but I don't think any of them are good enough to be effective for this case. Would you like to hear the 3 best ones I've thought of?"

It's a bad kind of shorthand to use. Stating that "these are the best solutions I know of right now, and I've exhausted my current ideas" is better than saying there's no solution - it avoids it seeming like you're just not interested.

Everything has a best solution after all, even if that solution may not be good enough.

This exactly.

Maybe a better way to say it is "There's no solution which I can implement in 5 minutes" or whatever time scale is suitable.

Sometimes the 'solution' isn't technical, but process/training changes, or other human-level modifications, not code or tech. People don't always like to hear that, but understanding that may be an option on the table, and what the implications are, can help decision makers make better decisions.

Good point. I won't say awareness of such solutions is a sign of a good programmer (though there's probably at least some correlation), but it's definitely the sign of a good consultant.