Question

在图中的五角星中，字母 $A, B, C, D$ 和 $E$ 分别被数字 $3, 5, 6, 7$ 和 $9$ 替换，但顺序不一定如此。线段 $AB, BC, CD, DE$ 和 $EA$ 两端数字的和形成一个等差数列，顺序不一定相同。这个等差数列的中间项是多少？

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 this 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 this order. What is the middle term of the arithmetic sequence?

$9$

$10$

$11$

$12$

$13$

答案：D

知识点：等差数列双重计数平均数

难度评级：1660

解答：

每个数字都是两条线段的端点，所以五个线段和的总和为 $2(3 + 5 + 6 + 7 + 9) = 60$ 。五项等差数列的中间项等于平均数，因此为 $\dfrac{60}{5} = 12$ 。

所以正确答案是 D。

Every number is an endpoint of two segments, so the five segment sums total $2(3 + 5 + 6 + 7 + 9) = 60.$ The middle term of a five-term arithmetic sequence equals its mean, which is $\dfrac{60}{5} = 12.$

Thus, the correct answer is D.

2005 AMC 10A 第 17 题

答案：D

难度评级：1660

解答：

其他年份的第 17 题