2005 AMC 10A 第 14 题

Question

有多少个三位数满足：中间数字是第一个数字和最后一个数字的平均数？

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

41

42

43

44

45

只有当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 share the same parity

分别数首位和末位都是奇数或都是偶数的数字对，注意首位不能为

0

。

Count first-and-last digit pairs that are both odd or both even, remembering the first digit cannot be

0

首位和末位数字必须奇偶性相同，平均数才是一个数字。若二者都为奇数，有

5 \cdot 5 = 25

对。若二者都为偶数，且首位非零，有

4 \cdot 5 = 20

对。每一对首末数字唯一确定中间数字，所以共有

25 + 20 = 45

个三位数。

所以正确答案是 E。

The first and last digits must have the same parity so their average is a digit. Both odd gives

5 \cdot 5 = 25

pairs. Both even, with a nonzero leading digit, gives

4 \cdot 5 = 20

pairs. Each pair fixes the middle digit, for a total of

25 + 20 = 45

numbers.

Thus, the correct answer is E.

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