Question

如图所示的五角星中，字母 $A$ 、 $B$ 、 $C$ 、 $D$ 、 $E$ 被数字 $3, 5, 6, 7$ 和 $9$ 替换，顺序不一定相同。线段 $AB$ 、 $BC$ 、 $CD$ 、 $DE$ 、 $EA$ 两端数字的和构成一个等差数列，顺序也不一定相同。这个等差数列的中项是多少？

In the five-sided star shown, the letters $A,$ $B,$ $C,$ $D,$ and $E$ are replaced by the numbers $3, 5, 6, 7,$ and $9,$ although not necessarily in that order. The sums of the numbers at the ends of the line segments $AB,$ $BC,$ $CD,$ $DE,$ and $EA$ form an arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence?

$9$

$10$

$11$

$12$

$13$

答案：D

知识点：等差数列双重计数

难度评级：1620

解答：

每个数字恰好作为五条线段中两条的端点，所以五个端点和的总和为 $2(3 + 5 + 6 + 7 + 9) = 60.$

五项等差数列的中项等于平均数，即 $60 \div 5 = 12$ 。

所以正确答案是 D。

Every number appears as an endpoint of exactly two of the five segments, so the total of the five sums is $2(3 + 5 + 6 + 7 + 9) = 60.$

The middle term of a five-term arithmetic sequence is its mean, namely $60 \div 5 = 12.$

Thus, the correct answer is D.

2005 AMC 12A 第 13 题

答案：D

难度评级：1620

解答：

其他年份的第 13 题