1963 AHSME Problem 32

The dimensions of a rectangle R are a and b, a < b. It is required to obtain a rectangle with dimensions x and y, x < a, y < 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:

\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \infty

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.