Element distinctness in O(n) time?
Element distinctness can be solved deterministically in the RAM model in time within $O(n\log\log n)\subset o(n\log n)$ time:

Sort in time within $O(n\log\log n)$ your $n$ numbers of $w$ bits using the sorting algorithm described by Han in STOC 2002 ("Deterministic sorting in $O(n\log\log n)$ time and linear space"), then scan in linear time for collisions.

As far as I know, that is the best result known to this day.