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This problem trains for: SAT-I, AMC-8, GMAT.
23 consecutive integers have an average of zero. Let k be any one of these integers. Which ones of the following statements are true?
The average of an odd number of consecutive integers is the number in the middle of the sequence, like this:
Obviously, if k is a number in this sequence, so is its opposite -k. Therefore, III is true.
Therefore, the middle number is 0 and the remaining 22 numbers are distributed evenly on both sides of the origin so as to cancel out. This means that any such number k satisfies:
This is equivalent to I, therefore I is true.
Adding 5 to both sides, we obtain V. Therefore V is true.
Adding 1 to both sides of II and then dividing by 2 we obtain:
which is obviously also true.