1974 AMC 12 Problem 30

Question

A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If

R

is the ratio of the lesser part to the greater part, then the value of

R^{\left[R^{\left(R^2+R^{-1}\right)}+R^{-1}\right]}+R^{-1}

is

2

2R

R^{-1}

2+R^{-1}

2+R

Translate the division condition into

R=\frac{1}{1+R}

From

R^2+R=1,

derive

R^{-1}=R+1

and simplify from the innermost exponent outward

Scale the greater part to

1,

making the lesser part

R.

The condition gives

R=\frac{1}{1+R},

so

R^2+R=1,\qquad R^{-1}=R+1.

Consequently

R^2+R^{-1}=R^2+R+1=2.

Working outward, the exponent of the outer

R

becomes

\begin{aligned} R^{\,R^2+R^{-1}}+R^{-1} &=R^2+R^{-1}\\ &=2. \end{aligned}

The whole expression is therefore

R^2+R^{-1}=2.

Therefore, the correct answer is A.

Problem 30 in other years