2000 AMC 10 Problem 24

Question

Let

f

be a function for which

f\left(\dfrac{x}{3}\right) = x^2 + x + 1.

Find the sum of all values of

z

for which

f(3z) = 7.

-\dfrac13

-\dfrac19

0

\dfrac59

\dfrac53

To make

\dfrac{x}{3} = 3z,

set

x = 9z

This gives a quadratic in

z;

use the sum-of-roots formula

-\dfrac{b}{a}

To evaluate

f(3z),

set

\dfrac{x}{3} = 3z,

so

x = 9z.

Then

\begin{aligned} f(3z) &= (9z)^2 + 9z + 1 \\ &= 81z^2 + 9z + 1. \end{aligned}

Setting this equal to

7

gives

81z^2 + 9z - 6 = 0.

By the sum-of-roots formula, the sum of the values of

z

is

-\dfrac{9}{81} = -\dfrac19.

Thus, the correct answer is B.

Problem 24 in other years