2005 AMC 12A Problem 11
Attempt Problem 11 of the 2005 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 12A solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
11.
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
Answer: E
Small hint:
The middle digit is an integer only when the first and last digits have the same parity
Big hint:
Count the odd-odd pairs and the even-even pairs separately, remembering the first digit is nonzero
Solution:
The middle digit is an integer only when the first and last digits are both odd or both even. Each such pair determines the middle digit uniquely.
There are odd-odd choices for the first and last digits. For even-even, the first digit cannot be giving choices.
The total is
Thus, the correct answer is E.
Problem 11 in other years
- 1950 AMC 12
- 1951 AMC 12
- 1952 AMC 12
- 1953 AMC 12
- 1954 AMC 12
- 1955 AMC 12
- 1956 AMC 12
- 1957 AMC 12
- 1958 AMC 12
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