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2005 AMC 10A Problem 14

Problem 14 of 25IntermediateNumber Theory

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?

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Solution

The first and last digits must have the same parity so their average is a digit. Both odd gives 5⋅5=255 \cdot 5 = 25 pairs. Both even, with a nonzero leading digit, gives 4⋅5=204 \cdot 5 = 20 pairs. Each pair fixes the middle digit, for a total of 25+20=4525 + 20 = 45 numbers. Thus, the correct answer is E.

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Concepts: digits · parity · casework

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.