1963 AMC 12 Problem 32

Question

The dimensions of a rectangle

R

are

a

and

b,

a\lt b.

It is required to obtain a rectangle with dimensions

x

and

y,

x\lt a,

y\lt a,

so that its perimeter is one-third that of

R,

and its area is one-third that of

R.

The number of such (different) rectangles is:

0

1

2

4

infinitely many

Translate the conditions into

3(x+y)=a+b

and

3xy=ab

Divide the sum equation by the product equation and compare reciprocals using

x\lt a,

y\lt a,

and

a\lt b

The conditions give

3(x+y)=a+b

and

3xy=ab.

Dividing yields

\frac1x+\frac1y=\frac1a+\frac1b.

But

x\lt a

and

y\lt a

make the left side greater than

\frac{2}{a},

while

a\lt b

makes the right side less than

\frac{2}{a}.

This is impossible.

Thus, the correct answer is A.

Problem 32 in other years