Bunuel wrote:
Is x an integer?
(1) The average (arithmetic mean) of x – 1 and x + 1 is an integer.
(2) The average (arithmetic mean) of x and –x is an integer.
(1) Average of x-1 and x+1 is = (x-1 + x+1)/2 = 2x/2 = x, which is given to be an integer. So x is an integer.
Sufficient.
(2) Average of x and -x is = (x - x)/2 = 0/2 = 0. Whatever be the value of x (integer or non-integer), sum of x and -x will always be 0, and thus average will also be 0. So we cant say whether x is an integer or not based on this information.
Insufficient.
Hence
A answer