noyoudaoicibaDictYouDict[no 词源字典]
"negative reply," early 13c., from Old English na (adv.) "no, never, not at all," from ne "not, no" + a "ever." First element from Proto-Germanic *ne (cognates: Old Norse, Old Frisian, Old High German ne, Gothic ni "not"), from PIE root *ne "no, not" (see un-). Second element from PIE *aiw- "vital force, life, long life, eternity" (see aye (adv.)).

As an adjective meaning "not any" (c. 1200) it is reduced from Old English nan (see none), the final -n omitted first before consonants and then altogether. As a noun from c. 1300. Phrase no can do "it is not possible" is attested from 1827, a locution of English-speaking Chinese noted 19c. in China, Australia and West Coast of U.S.
We repeated our advice again and again, but got no answer but a loud horse-laugh, and their national maxim of No can do: Europe fashion no do in China. ["Reminiscences of a Voyage to and from China," in "Paxton's Horticultural Register," London, 1836]
Construction no X, no Y attested from 1530s (in no peny no pardon). No problem as an interjection of assurance first attested 1963. No way as an expression meaning "it can't be done" is attested by 1968 (no way "by no means" is from c. 1400).[no etymology, no origin, 英语词源]
problem (n.)youdaoicibaDictYouDict
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" (see pro-) + ballein "to throw" (see ballistics).

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.
sweat (n.)youdaoicibaDictYouDict
Old English swat "perspiration, moisture exuded from the skin," also "labor, that which causes sweat," from Proto-Germanic *swaitaz "sweat" (cognates: Old Saxon, Old Frisian swet, Old Norse sveiti, Danish sved "sweat," Swedish svett, Middle Dutch sweet, Dutch zweet, Old High German sweiz, German Schweiß), from PIE *sweid- (2) "to sweat" (cognates: Sanskrit svedah "sweat," Avestan xvaeda- "sweat," Greek hidros "sweat, perspiration," Latin sudor, Lettish swiedri, Welsh chwys "sweat").

A widespread set of Slavic words (Polish, Russian pot "sweat") is from Old Church Slavonic potu, related to peku "heat," cognate with Latin coquere.

The Old English noun became Middle English swote, but later altered to the current form under the influence of the verb. Sweat of (one's) brow as a symbol of toil is from Gen. iii:19. Sweat equity is from 1968. Colloquial no sweat "no problem" attested from 1963.