problem has 2 distinct synonyms spread across 2 of its 3 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
a state of difficulty that needs to be resolved
“she and her husband are having problems”
a question raised for consideration or solution
“our homework consisted of ten problems to solve”
Synonyms, antonyms and the broader and narrower relations for problem come from Princeton University’s WordNet. Full sources and licences.