compete XVII (not frequent before XIX). — L. competere, in its late sense of ‘strive for (something) together with another’, f. COM- + petere aim at, seek.
So competition XVII. — late L. (cf. PETITION. competitive XIX. competitor fellow candidate, rival. XVI. — F. (-eur) or L.; see -OR 1.

com·pete / kəmˈpēt/ • v. [intr.] strive to gain or win something by defeating or establishing superiority over others who are trying to do the same: universities are competing for applicants. ∎  take part in a contest.