rivalry vs contest

rivalry and contest both are nouns.

Word	Noun	Adjective	Verb	Adverb
rivalry	Yes	No	No	No
contest	Yes	No	Yes	No

As nouns, contest is a hyponym of rivalry; that is, contest is a word with a more specific, narrower meaning than rivalry:

rivalry (noun)	contest (noun)
the act of competing as for profit or a prize	a struggle between rivals
	an occasion on which a winner is selected from among two or more contestants

rivalry (verb)	contest (verb)
	to make the subject of dispute, contention, or litigation