2005 AMC 12A 第 11 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
11.
有多少个三位数满足：中间数字是首位数字和末位数字的平均数？
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
小提示：
只有首位和末位数字奇偶性相同时，中间数字才是整数。
The middle digit is an integer only when the first and last digits have the same parity
大提示：
分别数奇奇配对和偶偶配对，注意首位数字不能是零。
Count the odd-odd pairs and the even-even pairs separately, remembering the first digit is nonzero
解答：
中间数字为整数，当且仅当首位数字与末位数字奇偶性相同。每一对这样的首末数字都唯一确定中间数字。
首末数字都是奇数的选择有 种；首末数字都是偶数时，首位不能为 ，所以有 种。
总数为 。
所以正确答案是 E。
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.
其他年份的第 11 题
- 1950 AMC 12
- 1951 AMC 12
- 1952 AMC 12
- 1953 AMC 12
- 1954 AMC 12
- 1955 AMC 12
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- 2002 AMC 12A
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- 2005 AMC 12B
- 2006 AMC 12A
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- 2009 AMC 12A
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- 2021 AMC 12A Spring
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