16.12 The method of dividing a line segment is to make the ratio of the smaller part to the larger part the same as the ratio of the larger part to the whole segment. If $R$ is the ratio of the smaller part to the larger part, then
$$
R^{\left[R^{\left(R^{2}+\frac{1}{R}\right)}+\frac{1}{R}\right]}+\frac{1}{R}
$$

is
(A) 2.
(B) $2 R$.
(C) $\frac{1}{R}$.
(D) $2+\frac{1}{R}$.
(E) $2+R$.
(25th American High School Mathematics Examination, 1974)
[Solution] Consider the line segment $AB$ divided by point $D$, and let $AD=x, DB=y$, with $y<x$, and $\frac{y}{x}=\frac{x}{x+y}$.

Since $\frac{y}{x}=R$, we can choose $x=1$, so $y=R$, and $R=\frac{1}{1+R}$, which means
$$
R^{2}+R-1=0 \text {, and } \quad \frac{1}{R}=R+1 \text {. }
$$

Thus, $R^{2}+\frac{1}{R}=R^{2}+R+1=\left(R^{2}+R-1\right)+2=2$,
so $R^{\left[R^{\left(R^{2}+\frac{1}{R}\right)}+\frac{1}{R}\right]}+\frac{1}{R}=R^{\left[R^{2}+\frac{1}{R}\right]}+\frac{1}{R}=R^{2}+\frac{1}{R}=2$. Therefore, the answer is $(A)$.