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chetan2u Hi Chetan, shouldn't the answer in the first part of the question be 'Yes'.
Since both median and mean are 80, it is clear that the 3rd value in a set of 5, would be 80.
Thus value(1) and/or value(2) could be less than 80, with value (3,4,5) being 80 and above.
So, we can successfully answer the first part of the question as 'no', given the above information.
Hence, the answer to the main question asked, should be 'Yes'.
­If we have 78,80,80,80,82, then S1 will be No
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chetan2u plz provide the official explanation­
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1. ­At Least 50% of the students scored more than 80 => No

It could be the case that all students score 80 except for the one scoring highest and the one scoring lowest, for example 82 and 78 respectively

Then there is only 1 (out of at least 3 people) scoring more than 80

2. There is at least one student who scored less than 80 => Yes

Range is 4
=> It cannot be the case that all students scored 80

Since median and mean = 80
=> At least 1 must score less than 80

3. No one scored 84 or more => Yes

If there is anyone scoring at least 84
=> then the one scoring lowest must score at least 80
=> Impossible since there must be at least 1 scoring less than 80