A little while back, I wanted to see if problem was related to probable. It is not, except for the “pro” prefix. I already went over probable’s origins (which, if you don't feel like clicking the link, are from the word prove) so today it’s time for problem.
Problem showed up in English in the late fourteenth century. As usual, it came from classical Latin (problema) by way of Old French(probléme). However in this case, the Romans got the word from the Greeks—problema or provlima if you want it phonetically. It came from proballein (phonetically provallan) a word meaning propose or put forward. The pro- part means forward, and the -ballein comes from throw or shot. A problem is an issue “thrown” forward, so it makes sense.
The part worth noting is that ballein is also the origin word for ballistics. It comes from the Latin ballista, an old word for a machine that throws stones (like a catapult), which in turn comes from the Greek ballistes, also from ballein. A ballistic is something shot, or thrown I guess. I’m sure anyone who’s ever been faced with a vexing dilemma can agree it can be like getting shot : ). Just look at calculus. Each problem is its own tiny bullet.