Etymology
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compete (v.)

1610s, " to enter or be put in rivalry with," from French compéter "be in rivalry with" (14c.), or directly from Late Latin competere "strive in common, strive after something in company with or together," in classical Latin "to meet or come together; agree or coincide; to be qualified," from com "with, together" (see com-) + petere "to strive, seek, fall upon, rush at, attack" (from PIE root *pet- "to rush, to fly").

According to OED, rare 17c., revived from late 18c. in sense "to strive (alongside another) for the attainment of something" and regarded early 19c. in Britain as a Scottish or American word. Market sense is from 1840s (perhaps a back-formation from competition); athletics sense attested by 1857. Intransitive use is by 1974. Related: Competed; competing.

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competency (n.)

1590s, "sufficiency to satisfy the wants of life," from Latin competentia "meeting together, agreement, symmetry," from competens, present participle of competere (see compete). Meaning "sufficiency of qualification" is recorded from 1797.

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competitive (adj.)

1826, "pertaining to or involving competition," from Latin competit-, past participle stem of competere (see compete) + -ive. Meaning "eager to compete, aggressive, ambitious" is by 1977. Related: Competitively; competitiveness.

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competitor (n.)

1530s, "one who competes in rivalry (with another), a rival," from French compétiteur (16c.), or directly from Latin competitor "rival," agent noun from past-participle stem of competere (see compete). As "one placed in a competition," 1650s.

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competent (adj.)

late 14c., "suitable, answering all requirements, sufficient, adequate," from Old French competent "sufficient, appropriate, suitable," and directly from Latin competentem (nominative competens), present participle of competere "coincide, agree" (see compete). It preserves the classical Latin sense of the verb, whereas the meaning in compete is a post-classical evolution. Meaning "able, fit, having ability or capacity" is from 1640s. Legal sense "having legal capacity or qualification" is late 15c. Related: Competently.

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competence (n.)

1590s, "rivalry" (based on compete), also "adequate supply," both senses now obsolete; 1630s as "sufficiency of means for living at ease," from French compétence, from Latin competentia "meeting together, agreement, symmetry," from competens, present participle of competere, especially in its earlier sense of "fall together, come together, be convenient or fitting" (see compete).

Meaning "adequate range of capacity or ability, sufficiency to deal with what is at hand" is from 1790. Legal sense "capability or fitness to be heard in court" is from 1708.

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competition (n.)

c. 1600, "action of seeking or endeavoring to gain what another is endeavoring to gain at the same time," from Late Latin competitionem (nominative competitio) "rivalry," in classical Latin "agreement," noun of action from past participle stem of competere (see compete).

Meaning "a contest for something, a trial of skill as a test of superiority or fitness" is from 1610s. Sense of "rivalry in the marketplace" attested from 1793; that of "entity or entities with which one competes" is from 1961, especially in business.

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*pet- 

Also petə-, Proto-Indo-European root meaning "to rush, to fly." 

It forms all or part of: accipiter; appetence; appetite; apterous; apteryx; archaeopteryx; asymptote; centripetal; Coleoptera; compete; competent; eurypterid; feather; helicopter; hippopotamus; Hymenoptera; impetigo; impetuous; impetus; iopterous; Lepidoptera; ornithopter; panache; panne; pen (n.1) "writing implement;" pennon; peripeteia; perpetual; perpetuity; petition; petulance; petulant; pin; pinion; pinnacle; pinnate; pinniped; potamo-; potamology; propitiation; propitious; ptero-; pterodactyl; ptomaine; ptosis; repeat; symptom.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit pattram "wing, feather, leaf," patara- "flying, fleeting;" Hittite pittar "wing;" Greek piptein "to fall," potamos "river, rushing water," pteron, pteryx "feather, wing," ptilon "soft feathers, down, plume;" Latin petere "to attack, assail; seek, strive after; ask for, beg; demand, require," penna "feather, wing;" Old Norse fjöðr, Old English feðer "feather;" Old Church Slavonic pero "feather;" Old Welsh eterin "bird."

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sumo (n.)
1880, from Japanese sumo "to compete."
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vie (v.)
1560s, "to bet, make a bet," (literally "make a vie, the noun attested from 1530s in cards), especially in card-playing, "to wager the value of one's hand against an opponent's," shortened form of Middle English envie "make a challenge," from Old French envier "compete (against), provoke; invite, summon, subpoena;" in gambling, "put down a stake, up the bet;" from Latin invitare "to invite," also "to summon, challenge" (see invitation). Sense of "to contend (with) in rivalry" in English is from 1560s; that of "to contend, compete, strive for superiority" is from c. 1600.
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