Question

A list contains n distinct integers. Are all n integers consecutive? (1) The average (arithmetic mean) of the list with the lowest number removed is 1 more than the average (arithmetic mean) of the list with the highest number removed. (2) The positive difference between any two numbers in the list is always less than n.

Answer 1

The list must contain consecutive integers. (They may not be arranged on the list in numerical order.)

Explanation:

Assume that L and H are the lowest and highest integers on the list, respectively. Assume that the sum of the other numbers on the list is X. The condition (1) tells us ...

(X + H)/(n -1) - (L + H)/(n -1) = 1

H - L = n -1 . . . . . . . . multiply by n-1 and simplify

In order for the highest and lowest of the n distinct integers to have a difference of n-1, the integers must be consecutive. Any pair will have a difference no less than 1 and no greater than n-1, meeting condition (2).

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Example:

Consider the 5 integers 13, 14, 15, 16, 17. The difference between highest and lowest is 17-13 = 4 = 5-1.

The average with the lowest removed is 15.5; the average with the highest removed is 14.5, which differs from the previous average by 1.