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Fifty candidates take a test in which they can get integer marks from zero to 100. If the mean mark is 60 find the maximum number of candidates who could have scored 100, the scores of the other candidates in this case and the standard deviation of the marks.

Show that if the mean is 60 and the standard deviation is 8 then it is impossible for 2 candidates or more to have scored 100 and if the standard deviation is 8.2 then at most 2 candidates could have scored 100. Find a set of marks for which 2 candidates get 100 and the mean is 60 and find the standard deviation in this case.

Show that if the standard deviation is $4\sqrt 2$ or less then it is impossible for anyone to have scored 100 marks.