Problem:
How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?
Answer Choices:
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Solution:
The set of the three digits of such a number can be arranged to form an increasing arithmetic sequence. There are 8 possible sequences with a common difference of 1 , since the first term can be any of the digits 0 through 7 . There are 6 possible sequences with a common difference of 2,4 with a common difference of 3 , and 2 with a common difference of 4 . Hence there are 20 possible arithmetic sequences. Each of the 4 sets that contain 0 can be arranged to form different numbers, and the 16 sets that do not contain 0 can be arranged to form different numbers. Thus there are a total of numbers with the required properties.
The problems on this page are the property of the MAA's American Mathematics Competitions