$20-31$ When the radius $r$ of a circle is increased by $n$, its area becomes 2 times, then $r$ equals
(A) $n(\sqrt{2}+1)$.
(B) $n(\sqrt{2}-1)$.
(C) $n$.
(D) $n(2-\sqrt{2})$.
(E) $\frac{n \pi}{\sqrt{2}+1}$.
(9th American High School Mathematics Examination, 1958)
[Solution] According to the problem, $\frac{\pi(r+n)^{2}}{\pi r^{2}}=2$, simplifying it yields $r^{2}-2 n r-n^{2}=0$, thus $r=(1+\sqrt{2}) n$. Therefore, the answer is $(A)$.