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Our team has won the tournament at only one time.

Our team has won the tournament at once.

Our team has won the tournament one time during tenure(since the team started).

Are these sentences correct grammatically? Is "tenure" correct here?

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Our team has won the tournament at only one time.

Our team has won the tournament at once. (At once means "immediately, right now, with no delay". e.g. Please come to the emergency assembly place at once. )

Our team has won the tournament one time since the team joined the league.

Is "tenure" correct here? No.

Tenure refers to a professor's position or an elected political position. It has nothing to do with a team's participating in a league. Please look in a good dictionary first.

https://www.macmillandictionary.com/dictionary/american/tenure

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No. Sayeg Our team has won the tournament only once.

eg Our team has won the tournament only once in its history.

Clive