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]