Full solution provided in attached file. By simply iterating through potential base sets for where all averages and range are equal we can find that there is a limited number of base sets that satisfy this condition. However as multiplying all values in the set by some positive integer maintains this property, there are infinitely many valid sets.
For the Cases A to F, I defined two lemmas which showcase how the median is affected by set size and which data values the median depends upon for sets of size 4, 5 and 6. The second lemma shows how repeating data values affect the three types of averages of the set.
From this Case A is satisfied by the set {1, 1,3,4,10} with 5 numbers.
Case B is shown in the full proof to be satisfiable with 4 numbers: {1, 1,4,10}.
The remaining cases are shows to be satisfiable with 6 numbers due to the additional degrees of freedom in value choices, for example case F being satisfied by the set {1, 2,3,3,10,10} and the other cases being solvable by similar constructions with reasons in the full reasoning in the file provided.
By Neel from Heckmondwike Grammar School