Olympic game scoring
Jelly, 12 bytes

ṢṖḊȯµÆmḤær0H


Try it online!

    µ           Take
Ṣ               the input sorted,
 Ṗ              without its last
  Ḋ             or first element,
   ȯ            or the unchanged input if that's empty,
     Æm         then calculate the mean,
       Ḥ        double it,
        ær      round it to the nearest multiple of
          0     10^-0 (= 1),
           H    and halve it.


A version which rounds halves down, in accordance with the spec at the expense of the second test case:

Jelly, 12 bytes

ṢṖḊȯµÆmḤ_.ĊH


Try it online!

The rounding method here is closer to Jonathan Allan's:

Ḥ        Double,
 _       subtract
  .      one half,
   Ċ     round up,
    H    and halve.