The difference between Contest and Fight
When used as nouns, contest means controversy, whereas fight means an occasion of fighting.
When used as verbs, contest means to contend, whereas fight means to contend in physical conflict, either singly or in war, battle etc.
check bellow for the other definitions of Contest and Fight
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Contest as a noun (uncountable):
Controversy; debate.
Examples:
"no contest"
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Contest as a noun (uncountable):
Struggle for superiority; combat.
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Contest as a noun (countable):
A competition.
Examples:
"The child entered the spelling contest."
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Contest as a verb (intransitive):
To contend.
Examples:
"I will contest for the open seat on the board."
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Contest as a verb (transitive):
To call into question; to oppose.
Examples:
"The rival contested the dictator's re-election because of claims of voting irregularities."
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Contest as a verb (transitive):
To strive earnestly to hold or maintain; to struggle to defend.
Examples:
"The troops contested every inch of ground."
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Contest as a verb (legal):
To make a subject of litigation; to defend, as a suit; to dispute or resist, as a claim, by course of law; to controvert.
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Fight as a verb (intransitive):
To contend in physical conflict, either singly or in war, battle etc.
Examples:
"A wounded animal will fight like a maniac, relentless, savage and murderous."
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Fight as a verb (reciprocal):
To contend in physical conflict with each other, either singly or in war, battle etc.
Examples:
"The two boxers have been fighting for more than half an hour."
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Fight as a verb (intransitive):
To strive for; to campaign or contend for success.
Examples:
"He fought for the Democrats in the last election."
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Fight as a verb (transitive):
To conduct or engage in (battle, warfare etc.).
Examples:
"The battle was fought just over that hill."
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Fight as a verb (transitive):
To engage in combat with; to oppose physically, to contest with.
Examples:
"My grandfather fought the Nazis in World War II."
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Fight as a verb (transitive):
To try to overpower; to fiercely counteract.
Examples:
"The government pledged to fight corruption."
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Fight as a verb (transitive, archaic):
To cause to fight; to manage or manoeuvre in a fight.
Examples:
"to fight cocks;  to fight one's ship"
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Fight as a noun:
An occasion of fighting.
Examples:
"One of them got stuck in a chokehold and got stabbed to death during the fight."
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Fight as a noun (archaic):
A battle between opposing armies.
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Fight as a noun:
A physical confrontation or combat between two or more people or groups.
Examples:
"Watch your language, are you looking for a fight?"
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Fight as a noun (sports):
A boxing or martial arts match.
Examples:
"I'm going to Nick’s to watch the big fight tomorrow night."
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Fight as a noun:
A conflict, possibly nonphysical, with opposing ideas or forces; strife.
Examples:
"I'll put up a fight to save this company."
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Fight as a noun (uncountable):
The will or ability to fight.
Examples:
"That little guy has a bit of fight in him after all. As soon as he saw the size of his opponent, all the fight went out of him."
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Fight as a noun (obsolete):
A screen for the combatants in ships.