Etymology
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problem (n.)

late 14c., probleme, "a difficult question proposed for discussion or solution; a riddle; a scientific topic for investigation," from Old French problème (14c.) and directly from Latin problema, from Greek problēma "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").

The meaning "a difficulty" is mid-15c. Mathematical sense of "proposition requiring some operation to be performed" is from 1560s in English. Problem child, one in which problems of a personal or social character are manifested, 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.

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problematic (adj.)

c. 1600, "doubtful, questionable, uncertain, unsettled," from French problematique (15c.), from Late Latin problematicus, from Greek problēmatikos "pertaining to a problem," from problēmatos, genitive of problēma (see problem).

Specific sense in logic, differentiating what is possible from what is necessarily true, is from 1610s. The sense of "constituting, containing, or causing a difficulty" is modern, probably from a noun use in sociology (1957). Related: Problematical (1560s); problematically.

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pro- 

word-forming element meaning "forward, forth, toward the front" (as in proclaim, proceed); "beforehand, in advance" (prohibit, provide); "taking care of" (procure); "in place of, on behalf of" (proconsul, pronoun); from Latin pro (adv., prep.) "on behalf of, in place of, before, for, in exchange for, just as," which also was used as a first element in compounds and had a collateral form por-.

Also in some cases from cognate Greek pro "before, in front of, sooner," which also was used in Greek as a prefix (as in problem). Both the Latin and Greek words are from PIE *pro- (source also of Sanskrit pra- "before, forward, forth;" Gothic faura "before," Old English fore "before, for, on account of," fram "forward, from;" Old Irish roar "enough"), extended form of root *per- (1) "forward," hence "in front of, before, toward, near," etc.

The common modern sense of "in favor of, favoring" (pro-independence, pro-fluoridation, pro-Soviet, etc.) was not in classical Latin and is attested in English from early 19c.

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*gwele- 

*gwelə-, also *gwel-, Proto-Indo-European root meaning "to throw, reach," with extended sense "to pierce."

It forms all or part of: anabolic; arbalest; astrobleme; ball (n.2) "dancing party;" ballad; ballet; ballista; ballistic; ballistics; belemnite; catabolism; devil; diabolical; discobolus; emblem; embolism; hyperbola; hyperbole; kill (v.); metabolism; palaver; parable; parabola; parley; parliament; parlor; parol; parole; problem; quell; quail (v.) "lose heart, shrink, cower;" symbol.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit apa-gurya "swinging," balbaliti "whirls, twirls;" Greek ballein "to throw, to throw so as to hit," also in a looser sense, "to put, place, lay," bole "a throw, beam, ray," belemnon "dart, javelin," belone "needle," ballizein "to dance;" Armenian kelem "I torture;" Old Church Slavonic zali "pain;" Lithuanian galas "end," gėla "agony," gelti "to sting."

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*per- (1)

Proto-Indo-European root forming prepositions, etc., meaning "forward," and, by extension, "in front of, before, first, chief, toward, near, against," etc.

It forms all or part of: afford; approach; appropriate; approve; approximate; barbican; before; deprive; expropriate; far; first; for; for-; fore; fore-; forefather; foremost; former (adj.); forth; frame; frau; fret; Freya; fro; froward; from; furnish; furniture; further; galore; hysteron-proteron; impervious; improbity; impromptu; improve; palfrey; par (prep.); para- (1) "alongside, beyond; altered; contrary; irregular, abnormal;" paradise; pardon; paramount; paramour; parvenu; pellucid; per; per-; percent; percussion; perennial; perestroika; perfect; perfidy; perform; perfume; perfunctory; perhaps; peri-; perish; perjury; permanent; permeate; permit; pernicious; perpendicular; perpetual; perplex; persecute; persevere; perspective; perspire; persuasion; pertain; peruse; pervade; pervert; pierce; portray; postprandial; prae-; Prakrit; pre-; premier; presbyter; Presbyterian; preterite; pride; priest; primal; primary; primate; primavera; prime; primeval; primitive; primo; primogenitor; primogeniture; primordial; primus; prince; principal; principle; prior; pristine; private; privilege; privy; pro (n.2) "a consideration or argument in favor;" pro-; probably; probe; probity; problem; proceed; proclaim; prodigal; produce; profane; profess; profile; profit; profound; profuse; project; promise; prompt; prone; proof; proper; property; propinquity; prophet; prose; prostate; prosthesis; protagonist; Protean; protect; protein; Proterozoic; protest; proto-; protocol; proton; protoplasm; Protozoa; proud; prove; proverb; provide; provoke; prow; prowess; proximate; Purana; purchase; purdah; reciprocal; rapprochement; reproach; reprove; veneer.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit pari "around, about, through," parah "farther, remote, ulterior," pura "formerly, before," pra- "before, forward, forth;" Avestan pairi- "around," paro "before;" Hittite para "outside of," Greek peri "around, about, near, beyond," pera "across, beyond," paros "before," para "from beside, beyond," pro "before;" Latin pro "before, for, on behalf of, instead of," porro "forward," prae "before," per "through;" Old Church Slavonic pra-dedu "great-grandfather;" Russian pere- "through;" Lithuanian per "through;" Old Irish ire "farther," roar "enough;" Gothic faura "before," Old English fore (prep.) "before, in front of," (adv.) "before, previously," fram "forward, from," feor "to a great distance, long ago;" German vor "before, in front of;" Old Irish air- Gothic fair-, German ver-, Old English fer-, intensive prefixes.

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brain-teaser (n.)
"difficult puzzle or problem," 1893, from brain (n.) + agent noun from tease (v.).
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headache (n.)
Old English heafodece; see head (n.) + ache (n.). Colloquial sense of "troublesome problem" is first recorded 1934. Related: Headachy (1705).
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drinking (n.)

late 12c., drinkinge, "the action of drinking," especially drinking for pleasure, verbal noun from drink (v.). Drinking problem "alcoholism" is by 1953; earlier was drinking habit (by 1825).

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solvitur ambulando 
an appeal to practical experience for a solution or proof, Latin, literally "(the problem) is solved by walking," originally in reference to the proof by Diogenes the Cynic of the possibility of motion.
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nitty-gritty (n.)

"basic facts of a situation or problem," by 1961, knitty-gritty, American English, said to have been chiefly used by black jazz musicians, perhaps ultimately from nit and grits "finely ground corn." As an adjective from 1966.

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