Show-off taking problem round university (6)
I believe the answer is:
poseur
'show-off' is the definition.
(poseur is a kind of show-off)
'problem round university' is the wordplay.
'problem' becomes 'poser' (poser is a kind of problem).
'round' indicates putting letters inside.
'university' becomes 'u'.
'poser' placed around 'u' is 'POSEUR'.
'taking' acts as a link.
(Other definitions for poseur that I've seen before include "One behaving affectedly to impress others" , "Person who's affected" , "Person who puts on airs, a phoney" , "One of hypocritical affectation" , "One who strikes attitudes, a phoney" .)