There's not just one big problem - there's millions (perhaps billions) of one big problems, depending on who's doing the asking and who's doing the answering. Sure, there are a few common problems that could be solved, but you have to figure out a way to ask the questions about the problems in such a way that the solution will be mutually agreeable to the largest number of people.
Energy, food, war, socks. All of these are big problems (OK, not so much with the socks) that won't be solved with one big solution. By the time you're done breaking them down into solvable units, some wierdo will come along and ask "So, what's the biggest problem?"
Solve that, and virtually everything in almost every industry changes.
* Mass killings of human beings by other human beings (whatever be the reason, whatever they are called)
* Mismanagement of global food inventory, as mentioned by others
* Inequality between women and men (and others). Existing solutions are not customized for cultures/communities.
It's mind boggling how uneven wealth distribution is in some countries.
It's the only thing that destroys everything it touches. And it often has massive reach and repercussions.