18.64 A rectangle $R$ has sides of length $a$ and $b$, where $a < b$. To obtain a rectangle with sides of length $x$ and $y$ (where $x < a$, $y < a$), such that its perimeter is $\frac{1}{3}$ of the perimeter of $R$, and its area is $\frac{1}{3}$ of the area of $R$, the number of different such rectangles is
(A) 0.
(B) 1.
(C) 2.
(D) 4.
(E) infinitely many.
(14th American High School Mathematics Examination, 1963)
[Solution] From the given, we have $3(x+y)=a+b$ and $3xy=ab$. Dividing both sides of the two equations, we get $\frac{1}{y}+\frac{1}{x}=\frac{1}{a}+\frac{1}{b}$. Since $x<a<b$ and $y<a<b$, the above equation cannot hold. Therefore, the answer is $(A)$.