close, equal (BrE), even, tight
closely fought (esp. BrE), hard-fought
one-sided, unequal (BrE)
The contest was too one-sided to be exciting.

open
This contest is wide open: any of half a dozen teams could win it.

fair
The other bidders for the contract complained that it had not been a fair contest.

exciting, good, great, real, thrilling
bitter
head-to-head
The contestants are eliminated one by one until the last two compete in a head-to-head contest.

global, international, local, national, regional
election, electoral, gubernatorial (AmE), leadership (esp. BrE), political, presidential, primary
beauty, popularity, song, talent
athletic (esp. AmE), sporting (esp. BrE), sports (esp. AmE)

have, hold, run
We have three major beauty contests a year.
Talent contests are held in the club.

sponsor (AmE)
compete in, enter, take part in
A third candidate has entered the contest for the Republican nomination.

lose, win

take place

during a/the contest
During the election contest newspapers are not allowed to publish public opinion polls.

in a/the contest
Both sides are predicting victory in this close contest.

out of a/the contest
Jackson has injured his knee and is now out of the contest.

contest against
He won his opening contest against Costa of Argentina.

contest between
the contest between these two great boxers

contest for
the contest for the leadership of the party

contest over
In the animal kingdom intruders usually lose contests over territories.

contest with
The New Zealanders are looking forward to future contests with South Africa.

a contest of skills (esp. AmE), a contest of strength
be no contest (= used when one side in a contest is much stronger than the other and is sure to win)
the winner of a contest