2018 AMC 12A Problem 4

Question

Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, “We are at least

6

miles away,” Bob replied, “We are at most

5

miles away.” Charlie then remarked, “Actually the nearest town is at most

4

miles away.” It turned out that none of the three statements was true. Let

d

be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of

d?

(0, 4)

(4, 5)

(4, 6)

(5, 6)

(5, \infty)

Each statement is false, so replace each with its negation

“At least

6

” false means

d \lt 6;

“at most

5

” false means

d \gt 5;

“at most

4

” false means

d \gt 4

Negating the three false statements gives

d \lt 6,

d \gt 5,

and

d \gt 4.

The intersection of these conditions is

5 \lt d \lt 6,

that is, the interval

(5, 6).

Thus, the correct answer is D.

Problem 4 in other years