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Do you accept both of them?

This university only admits students who have passed the exam with distinction.

This university only admits students passed the exam with distinction.
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A pet grievance I have with the position of "only". I would place "only" in front of "students". I realize that the meaning of the sentence is understood, but to me it looks like [all the university does is admit students, and then does nothing more: no teaching, no housing, etc.].

I've had this phenomenon described to me as something like "only-moving-left" syndrome, but I find it a bit annoying nonetheless.
Comments  
This university only admits students who have passed the exam with distinction. -- OK.

This university only admits students passed the exam with distinction. -- Incorrect.
Students: We have free audio pronunciation exercises.
How about this one?

This university only admits students having passed the exam with distinction.
kl004535This university only admits students having passed the exam with distinction.
No, this is awkward.
Thank you, Mr Wordy
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 Philip's reply was promoted to an answer.