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?

$41$

$42$

$43$

$44$

$45$

答案：E

知识点：数字奇偶性分类讨论

难度评级：1460

解答：

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.

2005 AMC 10A 第 14 题

答案：E

难度评级：1460

解答：

其他年份的第 14 题