Question

在这场 AMC 10 编写会议结束返程时，竞赛主席注意到他的机场停车收据上的数字形如 $bbcac$ ，其中 $0\le a\lt b\lt c\le 9$ ，且 $b$ 是 $a$ 和 $c$ 的平均数。有多少个不同的五位数满足所有这些性质？

On the trip home from the meeting where this AMC 10 was constructed, the Contest Chair noted that his airport parking receipt had digits of the form $bbcac,$ where $0\le a\lt b\lt c\le 9,$ and $b$ was the average of $a$ and $c.$ How many different five-digit numbers satisfy all these properties?

$12$

$16$

$18$

$20$

$24$

答案：D

知识点：数字奇偶性组合

难度评级：1290

解答：

一旦选定 $a$ 和 $c$ ， $b=\dfrac{a+c}{2}$ 就确定，且 $a\lt b\lt c$ 自动成立。为了使 $b$ 为整数， $a$ 和 $c$ 必须同奇偶。

从 $\{0,2,4,6,8\}$ 中选两个，有 $\binom{5}{2}=10$ 对；从 $\{1,3,5,7,9\}$ 中选两个，又有 $10$ 对。

因此共有 $20$ 个有效数字。

所以正确答案是 D。

Once $a$ and $c$ are chosen, $b=\dfrac{a+c}{2}$ is determined, and $a\lt b\lt c$ holds automatically. For $b$ to be an integer, $a$ and $c$ must share parity.

Choosing two even digits from $\{0,2,4,6,8\}$ gives $\binom{5}{2}=10$ pairs, and choosing two odd digits from $\{1,3,5,7,9\}$ gives another $10.$

This yields $20$ valid numbers.

Thus, the correct answer is D.

2007 AMC 10B 第 8 题

答案：D

难度评级：1290

解答：

其他年份的第 8 题