Etymology
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-head 
word-forming element meaning "state or condition of being," Middle English -hede, from a variant of Old English -had, the source of -hood. The only surviving words with it are maidenhead and godhead.
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running (adj.)

"that runs, capable of moving quickly," late 14c., rennynge, present-participle adjective from run (v.), replacing earlier erninde, from Old English eornende. The meaning "rapid, hasty, done on the run" is from c. 1300. The sense of "continuous, carried on continually" is from late 15c.

Running-jump is from 1914. A running-mate (1865) originally was a horse entered in a race to set the pace for another from the same stable who was intended to win; U.S. "vice-presidential candidate" sense is recorded from 1888. Running-board is attested by 1817 in reference to a narrow gangway on either side of a ship or boat; extended by 1907 to the footboards of cars and trucks. 

Running dog is recorded by 1937, from Chinese and later North Korean communist phrases used to describe supposed imperialist lackeys, such as Mandarin zou gou "running dog," on the notion of a dog that runs at its master's command.

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running (n.)

Old English ærning, "act of one who or that which runs, rapid motion on foot," verbal noun from run (v.). Of a ship, "the action of sailing," 1680s.

Colloquial phrases in (or out) of the running "among (or not among) the lead competitors, competing (or not competing) in a race" (1863) is a metaphor from horse racing, where make the running "set the pace" is recorded from 1837; hence "likely to succeed." Running-shoe is from 1884.

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head (v.)
"to be at the head or in the lead," c. 1200, from head (n.). Meaning "to direct the head (toward)" is from c. 1600. Related: headed, heading. The earliest use of the word as a verb meant "behead" (Old English heafdian). Verbal phrase head up "supervise, direct" is attested by 1930.
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head (adj.)
"most important, principal, leading," c. 1200, from head (n.). Old English heafod was used in this sense in compounds.
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head (n.)

Old English heafod "top of the body," also "upper end of a slope," also "chief person, leader, ruler; capital city," from Proto-Germanic *haubid (source also of Old Saxon hobid, Old Norse hofuð, Old Frisian haved, Middle Dutch hovet, Dutch hoofd, Old High German houbit, German Haupt, Gothic haubiþ "head"), from PIE root *kaput- "head."

Modern spelling is early 15c., representing what was then a long vowel (as in heat) and remained after pronunciation shifted. Of rounded tops of plants from late 14c. Meaning "origin of a river" is mid-14c. Meaning "obverse of a coin" (the side with the portrait) is from 1680s; meaning "foam on a mug of beer" is first attested 1540s; meaning "toilet" is from 1748, based on location of crew toilet in the bow (or head) of a ship.

Synechdochic use for "person" (as in head count) is first attested late 13c.; of cattle, etc., in this sense from 1510s. As a height measure of persons, from c. 1300. Meaning "drug addict" (usually in a compound with the preferred drug as the first element) is from 1911.

To be over (one's) head "beyond one's comprehension" is by 1620s. To give head "perform fellatio" is from 1950s. Phrase heads will roll "people will be punished" (1930) translates Adolf Hitler. Head case "eccentric or insane person" is from 1966. Head game "mental manipulation" attested by 1972.

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cat-head (n.)

"beam projecting from each side of the bows of a ship to hold the anchor away from the body of the ship," 1620s, from cat (n.) in some obscure sense.

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long-running (adj.)
1943, of theatrical productions, from long (adv.) + present participle of run (v.). Related: Longest-running.
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head-butt (n.)
also headbutt, 1935, from head (n.) + butt (n.5). As a verb, by 1946. Related: Head-butting (1917 as a noun).
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match-head (n.)

"piece of some chemical composition with which a match is tipped," 1860, from match (n.1) + head (n.).

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