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I know that run and run are past and present tense, although this is the sentence I see on some websites. Which one is correct and why?
As Stoney stated, this is a passive construction involving "be".
The passive is used to denote that the topic of the sentence is acted upon by something else, and the "something else" need not appear in the sentence.
The boy was attacked.
Exams are written at the end of the year.
The passive is constructed with the appropriate form of "be" followed by the past participle of the verb.
Since you need the past participle for the passive voice, "run" must be used as it's the participle we need. ("run" is an irregular verb, as you knew already.)
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