1961 AMC 12 Problem 24

Question

Thirty-one books are arranged from left to right in order of increasing prices. The price of each book differs by

\$2

from that of each adjacent book. For the price of the book at the extreme right a customer can buy the middle book and an adjacent one. Then:

the adjacent book referred to is at the left of the middle book

the middle book sells for

\$36

the cheapest book sells for

\$4

the most expensive book sells for

\$64

none of these are correct

Let the cheapest price be

p

; identify the

16

th and

31

st prices

Test the two books adjacent to the middle book and reject any case forcing a nonpositive price

The prices are

p,

p+2,

\ldots,

p+60,

and the middle price is

p+30.

If the right neighbor is used, then

(p+30)+(p+32)=p+60,

which gives

p=-2,

impossible for a selling price. The left neighbor gives

(p+30)+(p+28)=p+60,

so

p=2.

Thus the adjacent book must be the one to the left.

Therefore, the correct answer is A.

Problem 24 in other years