1982 AHSME Problems/Problem 28
Problem
A set of consecutive positive integers beginning with is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is . What number was erased?
Solution
Suppose that there are positive integers in the set initially, so their sum is by arithmetic series. The average of the remaining numbers is minimized when is erased, and is maximized when is erased.
It is clear that We write and solve a compound inequality for from which is either or
Let be the number that is erased. We are given that or
- If then becomes from which contradicting the precondition that is a positive integer.
- If then becomes from which
Remark
From note that the left side must be an integer, so must be the right side. It follows that is divisible by so
~MRENTHUSIASM
See Also
1982 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
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