1968 AMC 12 第 22 题

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

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

四段长度之和为

1

The four lengths sum to

1

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.

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