What is the main problem in the book The Hunger Games?
This is a broad question that invites many answers. However, if you are talking about only the main problem, I would argue that it is a problem of survival on two levels.
First, in this dystopian world, people called tributes have to fight to the death in an artificial arena against each other for the "viewing pleasure" of people who live in the capitol city. In the grand scheme of things there is a problem of social structure. The capitol is subjugating and oppressing all the districts socially, economically, and even physically. Hence, there is a huge conflict here.
On a more micro-level, the book deals with Katniss. She is the protagonist of the story who volunteers to take her sister's place as a tribute. During this time, she has to fight to survive. She endures hardships and deadly ordeals, which include new relationships and the need to gain help from others in order to survive. In this sense, she has to fight other tributes and even her own feelings.