late 14c., "a difficult question proposed for solution," from Old French problème (14c.) and directly from Latin problema, from Greek problema "a task, that which is proposed, a question;" also "anything projecting, headland, promontory; fence, barrier;" also "a problem in geometry," literally "thing put forward," from proballein "propose," from pro "forward" (from PIE root *per- (1) "forward") + ballein "to throw" (from PIE root *gwele- "to throw, reach").
Meaning "a difficulty" is mid-15c. Mathematical sense is from 1560s in English. Problem child is recorded by 1916. Phrase _______ problem in reference to a persistent and seemingly insoluble difficulty is attested from at least 1882, in Jewish problem. Response no problem "that is acceptable; that can be done without difficulty" is recorded from 1968.