1968 AMC 12 Problem 22

Question

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:

equal to

\dfrac14

equal to or greater than

\dfrac18

and less than

\dfrac12

greater than

\dfrac18

and less than

\dfrac12

greater than

\dfrac18

and less than

\dfrac14

less than

\dfrac12

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

The four lengths sum to

1

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.

Problem 22 in other years