There will always be problems, but for a certain kind of person, this is ideal; there will always be problems to solve.

Do you believe your apathy is an obstacle to solving problems? Or have you simply not found a problem you care enough about? Or do you mean that you feel overwhelmed by the number and severity of problems you perceive in the world? Or are you actually discouraged by a perceived inability to see solutions to problems? These are all different concerns, and while it isn't clear to me which (if any) of these are yours, I will try to address all of them.

As objectionable as some may find this, apathy (in the sense of being unable to summon deep concern for problems not affecting you directly) is not necessarily an obstacle to solving problems, but it certainly can be an obstacle to solving them well. You can care a great deal about the solution to a problem while caring little for the effect it has on others, but the less isolated the problem (the more people affected), the less likely it is your solution will solve the problem for everyone, or solve it adequately. Indeed, it is entirely possible to make a problem much worse or create a multitude of new problems without taking great care to avoid such things. There is some question as to whether it's possible to change anything whatsoever without doing some kind of violence in the world, and I advise you seek to minimize the violence you inflict however you can. Still, if you're genuinely concerned about an overabundance of apathy, know that a dose of compassion can help, and compassion can be cultivated. (On the subject of doing harm without meaning to, consider Zizek's Violence.)

Whatever your level of worldly concern, you'll still need motivation. What problems move you will be unique to your character and experiences, but I promise you: if nothing is appealing to you, you will need start working first and wait for motivation later. This may be disquieting and counter to intuition, but just as smiling triggers mirror neurons that can lead us to feel happy, so too can working on a problem cause us to care about it. By this, I don't necessarily mean toiling away at a keyboard in isolation; I mean you need to be steeped in the problem. Convince yourself the problem you perceive actually exists by meeting the people who experience it, first hand. Confuse yourself by meeting the people who don't. Understand how others have attempted to solve the problem. Too often you'll find the problem you perceived at the beginning was not a problem at all, and the real problem, if one even exists, is something else entirely. This is important work--far more important than crafting a solution--because you are working to find the right problem to solve.

Once you begin to comprehend the true depth and complexity of a problem, you can easily be overwhelmed by the many sub-problems and their intricacies. Here you must decide what you can do that would have the greatest impact, form a hypothesis regarding how it can be done, and begin iterating toward a solution in small steps. There is an absurd amount of literature on dealing with the many facets of such an enterprise, and I won't try to enumerate all that I've seen here, but I will urge you to exercise humility and generosity in whatever decisions you make. I rather liked the methodology presented in The Lean Startup, though it probably could have been condensed to an essay.

Finally, if you simply can't think of solutions, I suggest doing three things: solve other, smaller, simpler problems; read more about things related to the problem; read more about things prima facie unrelated to the problem. As for specific tactics, Polya's How to Solve It is never a bad place to start. Anything by Michael Michalko is probably worth reading, as well.

Whatever you decide, I wish you well in your endeavors!