There are nine meanings listed in OED's entry for the noun problem, three of which are labelled obsolete. See ‘Meaning & use’ for definitions, usage, and quotation evidence.
any question or matter involving doubt or difficulty: has financial and emotional problems. a statement requiring a solution, usually by means of mathematical operations: simple problems in addition. unruly: a problem child.
A problem is a question or puzzle that is intended to be solved or to be deeply thought about. Real-life examples: Your teacher may present a problem to the class so you can use what you’ve learned to solve it.
Difficulty in accepting or understanding or refusal to accept or understand. You made your best honest effort; if they judge you harshly, that’s their problem, not yours. (climbing) A set of moves required to complete a climb.
1. Difficult to deal with or control: a problem child. 2. Dealing with a moral or social problem: a problem play.