The Collaborative International Dictionary of English v.0.48:

Compete \Com*pete"\, v. i. [imp. & p. p. Competed; p. pr. &
   vb. n. Competing.] [L. completere, competitum; com- +
   petere to seek. See Petition.]
   To contend emulously; to seek or strive for the same thing,
   position, or reward for which another is striving; to contend
   in rivalry, as for a prize or in business; as, tradesmen
   compete with one another.
   [1913 Webster]

         The rival statesmen, with eyes fixed on America, were
         all the while competing for European alliances.
                                                  --Bancroft.
   [1913 Webster] Competence

WordNet (r) 3.0 (2006):

compete
    v 1: compete for something; engage in a contest; measure oneself
         against others [syn: compete, vie, contend]

Moby Thesaurus II by Grady Ward, 1.0:

52 Moby Thesaurus words for "compete":
   antagonize, approach, attempt, battle, beat against,
   beat up against, breast the wave, buck, buffet, buffet the waves,
   challenge, clash, close with, collide, combat, compete with,
   conflict, contend, contend against, contest, cope, dispute,
   emulate, equal, essay, fence, fight, fight against, grapple with,
   jockey, join battle with, joust, labor against, match, meet,
   militate against, offer resistance, outvie, reluct, reluctate,
   rival, stem the tide, strive, strive against, struggle,
   struggle against, take on, test one another, touch, try, vie,
   vie with