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

Answer: A

Concepts:rectangle inequality algebraic manipulation

Difficulty rating: 1990

Small Hint:

Translate the conditions into $3(x+y)=a+b$ and $3xy=ab$

Big Hint:

Divide the sum equation by the product equation and compare reciprocals using $x\lt a,$ $y\lt a,$ and $a\lt b$

Solution:

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.

1963 AMC 12 Problem 32

Answer: A

Difficulty rating: 1990

Small Hint:

Big Hint:

Solution:

Problem 32 in Other Years