Design solves problems.
Thus, as designers, we're little more than problem solvers.
I can't say I know where this sentiment comes from or have an exhaustive history of its use, but I've a sneaking suspicion that we're not the first to coin it.
I believe this because, while problem solvers we may be, there aren't many fields in which a need didn't arise to meet some sort of a problem.
Pipes are leaking. Here's a plumber.
House is on fire. This is a fireman.
Need, meet solution.
And while I'm aware that this notion is to trim the poetic fat from the title, it seems to be a view shared by enough of my peers and co-laborers that I've seen it in a number of talks. It's typically sandwiched somewhere between the introduction and a recommended course of action. The slide usually looks a little something like this.
At this point, I think it's safe to assume that most of us are aware that in one way or another, we're going to be solving some sort of problem--even those that haven't yet arisen. But I'd argue the prose-driven limitations applied to your title from the aforementioned don't actually add anything to the conversation.
Using a whole bunch of tape might stop up a broken dam. Using your hands to snuff out a fire could probably kill a blaze while also sufficiently marring your palms. Throwing a television out a window could successfully turn down the volume. But these solutions fail to understand the artistry in the decision-making process.
Design through the lens of problem-solving boils down what you do as little more than, at best, execution and, at worst, guesswork. Being defined by merely your end product neglects the merit of the process. It's a view that is measured only in results while design is employed for various reasons in a number of mediums and for a multitude of solutions. Some of which isn't measured in numbers, conversions, or bar graphs.
So, yes, designers are problem solvers. But in that same line of thought, so are plungers, hangers, napkins, tables, sporks, and the list goes on.
The thing about answers is that everyone's got them.