Question

将一条长为 $1$ 的线段分成四段。以这四段为边能组成四边形，当且仅当每一段都：

A segment of length $1$ is divided into four segments. Then there exists a quadrilateral with the four segments as sides if and only if each segment is:

等于 $\dfrac14$

equal to $\dfrac14$

大于或等于 $\dfrac18$ ，且小于 $\dfrac12$

equal to or greater than $\dfrac18$ and less than $\dfrac12$

大于 $\dfrac18$ ，且小于 $\dfrac12$

greater than $\dfrac18$ and less than $\dfrac12$

大于 $\dfrac18$ ，且小于 $\dfrac14$

greater than $\dfrac18$ and less than $\dfrac14$

小于 $\dfrac12$

less than $\dfrac12$

答案：E

知识点：三角不等式周长

难度评级：1610

小提示：

能组成非退化四边形，当且仅当最长边小于其余三边之和

A nondegenerate quadrilateral exists exactly when the longest side is shorter than the other three combined

大提示：

四段长度之和为 $1$

The four lengths sum to $1$

解答：

能组成简单非退化四边形，当且仅当每一边都小于其余三边之和。由于总长为 $1$ ，对每一段而言，这个条件是 $s\lt1-s$ ，即 $s\lt\frac{1}{2}$ 。

因此，正确答案是 E。

A simple nondegenerate quadrilateral exists exactly when each side is less than the sum of the other three. Since the total is $1,$ this condition is $s\lt1-s,$ or $s\lt\frac{1}{2},$ for every segment.

Therefore, the correct answer is E.

1968 AMC 12 第 22 题

答案：E

难度评级：1610

小提示：

大提示：

解答：

其他年份的第 22 题