problem (n.) Look up problem at Dictionary.com
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 first recorded 1920. Phrase _______ problem in reference to a persistent and seemingly insoluble difficulty is attested at least from 1882, in Jewish problem. Response no problem "that is acceptable; that can be done without difficulty" is recorded from 1968.